6.4: Levels of Policing and Role of Police
Learning Objectives
- Understand the various options for careers in the policing and law enforcement arena
- Discuss educational requirements are required for law enforcement positions with the federal government, state, county, municipal or city officer candidates
- Explain what a state police officer’s main objectives are
- Describe the difference between commissioned and civilian
- List several divisions that a commissioned officer can promote to
- List several departments that a civilian can work within a law enforcement type agency
- Discuss why different police departments work with each other
Critical Thinking Questions
- What education does a candidate need for jobs in federal law enforcement?
- What education does a candidate generally need for city or county jobs as a police officer?
- Is there a difference between a person who is considered commissioned and a person who is considered a civilian?
- Does every law enforcement agency have the same opportunities for advancement?
- Why do different police departments work together?

Policing Types
Federal Level: The federal arena of law enforcement careers is broad and vast. The options are almost endless, and the rewards outweigh most of the other local agencies. However, there is a catch, which centers on education and experience. Most law enforcement-related careers in the federal arena require a bachelor’s degree, at a minimum plus three years of related full-time work experience before applying.
- A bachelor’s degree in either accounting, computer science/information technology, foreign language (only a criminal justice major if the candidate is planning on working full-time for a law enforcement agency for at least three years before applying),
- OR a JD degree from an accredited law school,
- OR a diversified bachelor’s degree AND three years of professional experience, OR a master’s degree, or Ph.D. along with two years of professional experience.
Federal job possibilities (the list is not comprehensive)
- Federal Bureau of Investigation (FBI)
- Bureau of Alcohol, Tobacco, Firearms, and Explosives (ATF)
- Drug Enforcement Administration (DEA)
- Secret Service
- Central Intelligence Agency (CIA)
- National Security Agency (NSA)
- United States Marshals Service (USMS)
- U.S. Park Police
- U.S. Fish and Wildlife Service (USFWS)
- Department of Justice
- Federal Bureau of Prisons
- U.S. Army Criminal Investigation Command (CID)
- U.S. Army Counter Intelligence
- Dept. of Ariculture-Office of Inspector General (USDA-OIG)
- U.S. Forest Service Law Enforcement & Investigations (USFS LEI)
- Department of Commerce-Office of Inspector General (DOC-OIG)
- Office of Security (DOC-OS)
- US Commerce Department Police
- Office of Export Enforcement (OEE)
- National Institute of Standards and Technology Police
- United States Pentagon Police
- Department of Defense-Defense Criminal Investigative Service (DCIS)
- United States Pentagon Police (USPPD)
- Department of Defense Police
- Defense Security Police
- Defense Logistic Agency Police
- United States Coast Guard (USCG)
- Plus Many More ++
State Level: State police work for a state. For example, in Oregon, the state police work for the state of Oregon and the department is Oregon State Police (OSP). In Nevada, the state police are Nevada Highway Patrol (NHP) and work for the Nevada Department of Public Safety. The commissioned (certified by the state with powers of arrest) employees are generally called troopers (a.k.a. police officers), their uniform is blue (except for California Highway Patrol who wear tan and dark brown uniforms), they wear round tipped hats, and their primary duty is to patrol the highways and interstates; however, many state police also investigate many different criminal crimes, run criminal forensic labs, along with divisions of fish and wildlife. The duties of each state agency are different and unique to each state. For example, OSP has an explosives unit (see photo below).
County Level: There are 3,142 counties in the United States. [1] Each county has an elected Sheriff and deputies (a.k.a. police officers) work directly under the Sheriff. Deputies are no different from any other city police officer other than the Sheriff can be responsible for the courts and jails (a.k.a. correctional facility) in their respective county. But there are also many city police departments that are responsible for jails, so it just depends on the department.
- Jackson County Sheriff’s Office
- Josephine County Sheriff’s Office
- Douglas County Sheriff’s Office
- Deschutes County Sheriff’s Office
Municipal/City Level: Municipal/City police work under a municipality or city. If a city has a government, i.e., mayor and city council members and a municipal code (misdemeanor laws for the city), then the city can have city police. If a person works for a city, the designator is a police officer. Some cities have a connected jail (a.k.a. correction or detention facility), while others are operated through the Sheriff and the county.
- Ashland Police Department
- Talent Police Department
- Phoenix Police Department
- Medford Police Department
- Central Point Police Department
- Eagle Point Police Department
Miscellaneous Policing Jobs: There are many police jobs that may fall under the jurisdiction of the federal government, state, county, or city or they are civilian positions (see below):
- Bailiff for a Court
- Animal Control or Animal Cruelty Investigator
- Computer Forensics
- Correctional Counselor
- Court Clerk or Court Reporter
- Criminologist
- Private Investigator
- Criminal Justice Administration
- Crime Prevention Specialist
- Protection Officer
- Forensic Accountant, Anthropoligist, Artist, Hypnotists, Nurse, Pathologist, Psychologist, Scientist, Serologist, Toxicologist
- Judge
- Juvenile Probation Officer
- Latent Print Examiner
- Legal Secretary/Paralegal
- Loss Prevention Officer
- Mediator/Negotiator
- Pre-trial Officer
- Security Analyst
- Security Officer
- Social Worker
- Victims Advocate
- Plus Many More ++
Becoming an Officer Example
Divisions within Each Agency: Law enforcement agencies, whether they are federal, county, state, or municipal/city, generally have jobs available within two major areas: 1- Commissioned 2- Civilian. Commissioned is a term that describes an employee that has been through police training is certified as a police officer and has arresting powers in the state. Civilian is a term that describes an employee who has not been through police training and does not have arresting powers.
COMMISSIONED- Divisions within a Law Enforcement Agency:
- Detective/Investigations (Persons Crimes- Property Crimes-Homicide, Rape, Robbery, Burglary, Auto Theft, DUII, Domestic Violence)
- Motors
- Narcotics
- Human/Sex Trafficking
- VICE
- NCrime Scene Investigation (CSI)
- SWAT
- K-9 (patrol, drug, and search & rescue dogs)
- Crisis Negotiator
- Mounted Unit (horses)
- Air Unit
- Training/Range Master
- Academy/Tac Officer
- Bike Patrol
- Recruiting
- Internal Affairs
- Public Information Officer
- Gangs
- Search & Rescue
- Forest and Fish & Wildlife
- Marine
- Various Area Task Force (usually made up of various law enforcement agencies in the area- to sometimes include federal agencies too)
- Plus Many More ++
CIVILIAN- Divisions within a Law Enforcement Agency:
- Dispatch/911 Operator
- Records
- Crime Analysis
- Forensic Unit/CSI
- Training
- Fleet Management
- Support/Facilities
- Human Resources
- Operations Support Unit
- Recruitment Coordinator
- Volunteer Coordinator
- Administrative Support
- Plus Many More ++
Contact with Outside Agencies: It takes a team to accomplish policing. One federal, state, county, or city police agency cannot do it all alone. In order to succeed the agencies must work together. Whether a narcotics division works with the ATF or an entire SWAT team comprised of officers from various city and county departments, the team concept in policing is unwavering.
Medford Police, Ashland Police, Central Point Police, Jackson County Sheriff’s Office, Oregon State Police, Jackson County Community Corrections, Federal Bureau of Investigation, Homeland Security Investigations Division, and the Oregon Army National Guard. Not only does the job get done more efficiently and more effectively but also the communication that occurs when nine different agencies converge is awe-inspiring.
Finally, once one enters the policing field, the contacts made daily are numerous.
- Jails/Correction/Detention Facilities and Employees
- Prisons
- Prosecutors Office
- Defense Attorneys
- Judges and Lawyers
- Various Social Services
- Educational Entities
-
U.S. Department of the Interior (2018, September 23). How many countries are there in the United States. Retrieved from
https://www.usgs.gov/faqs/how-many-counties-are-there-united-states
.
↵


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acid
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substance that produces H
3
O
+
when dissolved in water
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acid-base reaction
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reaction involving the transfer of a hydrogen ion between reactant species
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actual yield
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amount of product formed in a reaction
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analyte
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chemical species of interest
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balanced equation
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chemical equation with equal numbers of atoms for each element in the reactant and product
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base
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substance that produces OH
−
when dissolved in water
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buret
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device used for the precise delivery of variable liquid volumes, such as in a titration analysis
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chemical equation
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symbolic representation of a chemical reaction
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coefficient
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number placed in front of symbols or formulas in a chemical equation to indicate their relative amount
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combustion analysis
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gravimetric technique used to determine the elemental composition of a compound via the collection and weighing of its gaseous combustion products
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combustion reaction
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vigorous redox reaction producing significant amounts of energy in the form of heat and, sometimes, light
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complete ionic equation
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chemical equation in which all dissolved ionic reactants and products, including spectator ions, are explicitly represented by formulas for their dissociated ions
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end point
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measured volume of titrant solution that yields the change in sample solution appearance or other property expected for stoichiometric equivalence (see
equivalence point
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equivalence point
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volume of titrant solution required to react completely with the analyte in a titration analysis; provides a stoichiometric amount of titrant for the sample’s analyte according to the titration reaction
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excess reactant
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reactant present in an amount greater than required by the reaction stoichiometry
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gravimetric analysis
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quantitative chemical analysis method involving the separation of an analyte from a sample by a physical or chemical process and subsequent mass measurements of the analyte, reaction product, and/or sample
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half-reaction
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an equation that shows whether each reactant loses or gains electrons in a reaction.
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indicator
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substance added to the sample in a titration analysis to permit visual detection of the end point
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insoluble
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of relatively low solubility; dissolving only to a slight extent
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limiting reactant
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reactant present in an amount lower than required by the reaction stoichiometry, thus limiting the amount of product generated
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molecular equation
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chemical equation in which all reactants and products are represented as neutral substances
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net ionic equation
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chemical equation in which only those dissolved ionic reactants and products that undergo a chemical or physical change are represented (excludes spectator ions)
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neutralization reaction
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reaction between an acid and a base to produce salt and water
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oxidation
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process in which an element’s oxidation number is increased by loss of electrons
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oxidation number
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(also, oxidation state) the charge each atom of an element would have in a compound if the compound were ionic
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oxidation-reduction reaction
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(also, redox reaction) reaction involving a change in oxidation number for one or more reactant elements
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oxidizing agent
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(also, oxidant) substance that brings about the oxidation of another substance, and in the process becomes reduced
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percent yield
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measure of the efficiency of a reaction, expressed as a percentage of the theoretical yield
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precipitate
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insoluble product that forms from reaction of soluble reactants
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precipitation reaction
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reaction that produces one or more insoluble products; when reactants are ionic compounds, sometimes called double-displacement or metathesis
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product
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substance formed by a chemical or physical change; shown on the right side of the arrow in a chemical equation
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quantitative analysis
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the determination of the amount or concentration of a substance in a sample
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reactant
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substance undergoing a chemical or physical change; shown on the left side of the arrow in a chemical equation
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reducing agent
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(also, reductant) substance that brings about the reduction of another substance, and in the process becomes oxidized
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reduction
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process in which an element’s oxidation number is decreased by gain of electrons
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salt
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ionic compound that can be formed by the reaction of an acid with a base that contains a cation and an anion other than hydroxide or oxide
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single-displacement reaction
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(also, replacement) redox reaction involving the oxidation of an elemental substance by an ionic species
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solubility
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the extent to which a substance may be dissolved in water, or any solvent
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soluble
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of relatively high solubility; dissolving to a relatively large extent
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spectator ion
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ion that does not undergo a chemical or physical change during a reaction, but its presence is required to maintain charge neutrality
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stoichiometric factor
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ratio of coefficients in a balanced chemical equation, used in computations relating amounts of reactants and products
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stoichiometry
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relationships between the amounts of reactants and products of a chemical reaction
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strong acid
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acid that reacts completely when dissolved in water to yield hydronium ions
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strong base
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base that reacts completely when dissolved in water to yield hydroxide ions
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theoretical yield
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amount of product that may be produced from a given amount of reactant(s) according to the reaction stoichiometry
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titrant
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solution containing a known concentration of substance that will react with the analyte in a titration analysis
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titration analysis
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quantitative chemical analysis method that involves measuring the volume of a reactant solution required to completely react with the analyte in a sample
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weak acid
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acid that reacts only to a slight extent when dissolved in water to yield hydronium ions
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weak base
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base that reacts only to a slight extent when dissolved in water to yield hydroxide ions
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4.12: Sequences and Series of Functions
This page is a draft and is under active development.
I. Let
\[f_{1}, f_{2}, \ldots, f_{m}, \dots\]
be a sequence of mappings from a common domain \(A\) into a metric space \(\left(T, \rho^{\prime}\right) .\) For each (fixed) \(x \in A,\) the function values
\[f_{1}(x), f_{2}(x), \ldots, f_{m}(x), \ldots\]
form a sequence of points in the range space \(\left(T, \rho^{\prime}\right).\) Suppose this sequence converges for each \(x\) in a set \(B \subseteq A.\) Then we can define a function \(f : B \rightarrow T\) by setting
\[f(x)=\lim _{m \rightarrow \infty} f_{m}(x) \text { for all } x \in B.\]
This means that
\[(\forall \varepsilon>0)(\forall x \in B)(\exists k)(\forall m>k) \quad \rho^{\prime}\left(f_{m}(x), f(x)\right)<\varepsilon.\]
Here \(k\) depends not only on \(\varepsilon\) but also on \(x,\) since each \(x\) yields a different sequence \(\left\{f_{m}(x)\right\}.\) However, in some cases (resembling uniform continuity), \(k\) depends on \(\varepsilon\) only; i.e., given \(\varepsilon>0,\) one and the same \(k\) fits all \(x\) in \(B.\) In symbols, this is indicated by changing the order of quantifiers, namely,
\[(\forall \varepsilon>0)(\exists k)(\forall x \in B)(\forall m>k) \quad \rho^{\prime}\left(f_{m}(x), f(x)\right)<\varepsilon.\]
Of course, (2) implies (1) , but the converse fails (see examples below). This suggests the following definitions.
With the above notation, we call \(f\) the pointwise limit of a sequence of functions \(f_{m}\) on a set \(B(B \subseteq A)\) iff
\[f(x)=\lim _{m \rightarrow \infty} f_{m}(x) \text { for all } x \text { in } B;\]
i.e., formula (1) holds. We then write
\[f_{m} \rightarrow f(\text {pointwise}) \text { on } B.\]
In case (2) , we call the limit uniform (on \(B )\) and write
\[f_{m} \rightarrow f(\text {uniformly}) \text { on } B.\]
II. If the \(f_{m}\) are real, complex, or vector valued (§3), we can also define \(s_{m}=\sum_{k=1}^{m} f_{k}\) (= sum of the first \(m\) functions) for each \(m\), so
\[(\forall x \in A)(\forall m) \quad s_{m}(x)=\sum_{k=1}^{m} f_{k}(x).\]
The \(s_{m}\) form a new sequence of functions on \(A.\) The pair of sequences
\[\left(\left\{f_{m}\right\},\left\{s_{m}\right\}\right)\]
is called the (infinite) series with general term \(f_{m} ; s_{m}\) is called its \(m\) th partial sum. The series is often denoted by symbols like \(\sum f_{m}, \sum f_{m}(x),\) etc.
The series \(\sum f_{m}\) on \(A\) is said to converge (pointwise or uniformly) to a function \(f\) on a set \(B \subseteq A\) iff the sequence \(\left\{s_{m}\right\}\) of its partial sums does as well.
We then call \(f\) the sum of the series and write
\[f(x)=\sum_{k=1}^{\infty} f_{k}(x) \text { or } f=\sum_{m=1}^{\infty} f_{m}=\lim s_{m}\]
(pointwise or uniformly) on \(B\).
Note that series of constants, \(\sum c_{m},\) may be treated as series of constant functions \(f_{m},\) with \(f_{m}(x)=c_{m}\) for \(x \in A.\)
If the range space is \(E^{1}\) or \(E^{*},\) we also consider infinite limits,
\[\lim _{m \rightarrow \infty} f_{m}(x)=\pm \infty.\]
However, a series for which
\[\sum_{m=1}^{\infty} f_{m}=\lim s_{m}\]
is infinite for some \(x\) is regarded as divergent (i.e., not convergent) at that \(x\).
III. Since convergence of series reduces to that of sequences \(\left\{s_{m}\right\},\) we shall first of all consider sequences. The following is a simple and useful test for uniform convergence of sequences \(f_{m} : A \rightarrow\left(T, \rho^{\prime}\right).\)
Given a sequence of functions \(f_{m} : A \rightarrow\left(T, \rho^{\prime}\right),\) let \(B \subseteq A\) and
\[Q_{m}=\sup _{x \in B} \rho^{\prime}\left(f_{m}(x), f(x)\right).\]
Then \(f_{m} \rightarrow f(\text {uniformly on } B)\) iff \(Q_{m} \rightarrow 0\).
- Proof
-
If \(Q_{m} \rightarrow 0,\) then by definition
\[(\forall \varepsilon>0)(\exists k)(\forall m>k) \quad Q_{m}<\varepsilon.\]
However, \(Q_{m}\) is an upper bound of all distances \(\rho^{\prime}\left(f_{m}(x), f(x)\right), x \in B.\) Hence (2) follows.
Conversely, if
\[(\forall x \in B) \quad \rho^{\prime}\left(f_{m}(x), f(x)\right)<\varepsilon,\]
then
\[\varepsilon \geq \sup _{x \in B} \rho^{\prime}\left(f_{m}(x), f(x)\right),\]
i.e., \(Q_{m} \leq \varepsilon.\) Thus (2) implies
\[(\forall \varepsilon>0)(\exists k)(\forall m>k) \quad Q_{m} \leq \varepsilon\]
and \(Q_{m} \rightarrow 0.\) \(\square\)
(a) We have
\[\lim _{n \rightarrow \infty} x^{n}=0 \text { if }|x|<1 \text { and } \lim _{n \rightarrow \infty} x^{n}=1 \text { if } x=1.\]
Thus, setting \(f_{n}(x)=x^{n},\) consider \(B=[0,1]\) and \(C=[0,1)\).
We have \(f_{n} \rightarrow 0\) (pointwise) on \(C\) and \(f_{n} \rightarrow f(\text { pointwise })\) on \(B,\) with \(f(x)=0\) for \(x \in C\) and \(f(1)=1.\) However, the limit is not uniform on \(C,\) let alone on \(B .\) Indeed,
\[Q_{n}=\sup _{x \in C}\left|f_{n}(x)-f(x)\right|=1 \text { for each } n.\]
Thus \(Q_{n}\) does not tend to \(0,\) and uniform convergence fails by Theorem 1.
(b) In Example (a), let \(D=[0, a], 0<a<1 .\) Then \(f_{n} \rightarrow f\) (uniformly) on \(D\) because, in this case,
\[Q_{n}=\sup _{x \in D}\left|f_{n}(x)-f(x)\right|=\sup _{x \in D}\left|x^{n}-0\right|=a^{n} \rightarrow 0.\]
(c) Let
\[f_{n}(x)=x^{2}+\frac{\sin n x}{n}, \quad x \in E^{1}.\]
For a fixed \(x\),
\[\lim _{n \rightarrow \infty} f_{n}(x)=x^{2} \quad \text { since }\left|\frac{\sin n x}{n}\right| \leq \frac{1}{n} \rightarrow 0.\]
Thus, setting \(f(x)=x^{2},\) we have \(f_{n} \rightarrow f\) (pointwise) on \(E^{1}.\) Also,
\[\left|f_{n}(x)-f(x)\right|=\left|\frac{\sin n x}{n}\right| \leq \frac{1}{n}.\]
Thus \((\forall n) Q_{n} \leq \frac{1}{n} \rightarrow 0.\) By Theorem 1, the limit is uniform on all of \(E^{1}.\)
Let \(f_{m} : A \rightarrow\left(T, \rho^{\prime}\right)\) be a sequence of functions on \(A \subseteq(S, \rho).\) If \(f_{m} \rightarrow f\left(\text {uniformly } \text { on a set } B \subseteq A, \text { and if the } f_{m} \text { are relatively (or uniformly) }\right.\) continuous on \(B\), then the limit function \(f\) has the same property.
- Proof
-
Fix \(\varepsilon>0.\) As \(f_{m} \rightarrow f\) (uniformly) on \(B,\) there is a \(k\) such that
\[(\forall x \in B)(\forall m \geq k) \quad \rho^{\prime}\left(f_{m}(x), f(x)\right)<\frac{\varepsilon}{4}.\]
Take any \(f_{m}\) with \(m>k,\) and take any \(p \in B.\) By continuity, there is \(\delta>0,\) with
\[\left(\forall x \in B \cap G_{p}(\delta)\right) \quad \rho^{\prime}\left(f_{m}(x), f_{m}(p)\right)<\frac{\varepsilon}{4}.\]
Also, setting \(x=p\) in (3) gives \(\rho^{\prime}\left(f_{m}(p), f(p)\right)<\frac{\varepsilon}{4}.\) Combining this with (4) and (3), we obtain \(\left(\forall x \in B \cap G_{p}(\delta)\right)\)
\[\begin{aligned} \rho^{\prime}(f(x), f(p)) & \leq \rho^{\prime}\left(f(x), f_{m}(x)\right)+\rho^{\prime}\left(f_{m}(x), f_{m}(p)\right)+\rho^{\prime}\left(f_{m}(p), f(p)\right) \\ &<\frac{\varepsilon}{4}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}<\varepsilon. \end{aligned}\]
We thus see that for \(p \in B\),
\[(\forall \varepsilon>0)(\exists \delta>0)\left(\forall x \in B \cap G_{p}(\delta)\right) \quad \rho^{\prime}(f(x), f(p))<\varepsilon,\]
i.e., \(f\) is relatively continuous at \(p(\text { over } B),\) as claimed.
Quite similarly, the reader will show that \(f\) is uniformly continuous if the \(f_{n}\) are. \(\square\)
Note 2. A similar proof also shows that if \(f_{m} \rightarrow f\) (uniformly) on \(B,\) and if the \(f_{m}\) are relatively continuous at a point \(p \in B,\) so also is \(f.\)
Let \(\left(T, \rho^{\prime}\right)\) be complete. Then a sequence \(f_{m} : A \rightarrow T, A \subseteq(S, \rho),\) converges uniformly on a set \(B \subseteq A\) iff
\[(\forall \varepsilon>0)(\exists k)(\forall x \in B)(\forall m, n>k) \quad \rho^{\prime}\left(f_{m}(x), f_{n}(x)\right)<\varepsilon.\]
- Proof
-
If (5) holds then, for any (fixed) \(x \in B,\left\{f_{m}(x)\right\}\) is a Cauchy sequence of points in \(T,\) so by the assumed completeness of \(T,\) it has a limit \(f(x).\) Thus we can define a function \(f : B \rightarrow T\) with
\[f(x)=\lim _{m \rightarrow \infty} f_{m}(x) \text { on } B.\]
To show that \(f_{m} \rightarrow f\) (uniformly) on \(B,\) we use (5) again . Keeping \(\varepsilon, k,\) \(x,\) and \(m\) temporarily fixed, we let \(n \rightarrow \infty\) so that \(f_{n}(x) \rightarrow f(x)\). Then by Theorem 4 of Chapter 3, §15, \(\rho^{\prime}\left(f_{m}(x), f_{n}(x)\right) \rightarrow p^{\prime}\left(f(x), f_{m}(x)\right).\) Passing to the l imit in (5), we thus obtain (2) .
The easy proof of the converse is left to the reader (cf. Chapter 3, §17, Theorem 1). \(\square\)
IV. If the range space \(\left(T, \rho^{\prime}\right)\) is \(E^{1}, C,\) or \(E^{n}\) (*or another normed space), the standard metric applies. In particular, for series we have
\[\begin{aligned} \rho^{\prime}\left(s_{m}(x), s_{n}(x)\right) &=\left|s_{n}(x)-s_{m}(x)\right| \\ &=\left|\sum_{k=1}^{n} f_{k}(x)-\sum_{k=1}^{m} f_{k}(x)\right| \\ &=\left|\sum_{k=m+1}^{n} f_{k}(x)\right| \quad \text { for } m<n. \end{aligned}\]
Replacing here \(m\) by \(m-1\) and applying Theorem 3 to the sequence \(\left\{s_{m}\right\},\) we obtain the following result.
Let the range space of \(f_{m}, m=1,2, \ldots,\) be \(E^{1}, C,\) or \(E^{n}\) (*or another complete normed space). Then the series \(\sum f_{m}\) converges uniformly on \(B\) iff
\[(\forall \varepsilon>0)(\exists q)(\forall n>m>q)(\forall x \in B) \quad\left|\sum_{k=m}^{n} f_{k}(x)\right|<\varepsilon.\]
Similarly, via \(\left\{s_{m}\right\},\) Theorem 2 extends to series of functions. (Observe that the \(s_{m}\) are continuous if the \(f_{m}\) are.) Formulate it!
V. If \(\sum_{m=1}^{\infty} f_{m}\) exists on \(B,\) one may arbitrarily "group" the terms, i.e., replace every several consecutive terms by their sum. This property is stated more precisely in the following theorem.
Let
\[f=\sum_{m=1}^{\infty} f_{m}(\text {pointwise}) \text { on } B.\]
Let \(m_{1}<m_{2}<\cdots<m_{n}<\cdots\) in \(N,\) and define
\[g_{1}=s_{m_{1}}, \quad g_{n}=s_{m_{n}}-s_{m_{n-1}}, \quad n>1.\]
(Thus \(g_{n+1}=f_{m_{n}+1}+\cdots+f_{m_{n+1}}.)\) Then
\[f=\sum_{n=1}^{\infty} g_{n}(\text {pointwise}) \text { on } B \text { as well; }\]
similarly for uniform convergence.
- Proof
-
Let
\[s_{n}^{\prime}=\sum_{k=1}^{n} g_{k}, \quad n=1,2, \ldots\]
Then \(s_{n}^{\prime}=s_{m_{n}}\) (verify!), so \(\left\{s_{n}^{\prime}\right\}\) is a subsequence, \(\left\{s_{m_{n}}\right\},\) of \(\left\{s_{m}\right\} .\) Hence \(s_{m} \rightarrow f(\text { pointwise })\) implies \(s_{n}^{\prime} \rightarrow f\) (pointwise); i.e.,
\[f=\sum_{n=1}^{\infty} g_{n} \text { (pointwise). }\]
For uniform convergence, see Problem 13 (cf. also Problem 19). \(\square\)


3.2: Terminology
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Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An experiment is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a chance experiment. Flipping one fair coin twice is an example of an experiment.
A result of an experiment is called an outcome . The sample space of an experiment is the set of all possible outcomes. Three ways to represent a sample space are: to list the possible outcomes, to create a tree diagram , or to create a Venn diagram . The uppercase letter S is used to denote the sample space. For example, if you flip one fair coin, \(S = \{\text{H, T}\}\) where \(\text{H} =\) heads and \(\text{T} =\) tails are the outcomes.
An event is any combination of outcomes. Upper case letters like \(\text{A}\) and \(\text{B}\) represent events. For example, if the experiment is to flip one fair coin, event \(\text{A}\) might be getting at most one head. The probability of an event \(\text{A}\) is written \(P(\text{A})\).
Definition: Probability
The probability of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive (that is, zero and one and all numbers between these values).
- \(P(\text{A}) = 0\) means the event \(\text{A}\) can never happen.
- \(P(\text{A}) = 1\) means the event \(\text{A}\) always happens.
- \(P(\text{A}) = 0.5\) means the event \(\text{A}\) is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches 0.5 (the probability of heads).
Equally likely means that each outcome of an experiment occurs with equal probability. For example, if you toss a fair , six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (\(\text{H}\)) and a Tail (\(\text{T}\)) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.
To calculate the probability of an event A when all outcomes in the sample space are equally likely , count the number of outcomes for event \(\text{A}\) and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is \(\{\text{HH, TH, HT,TT}\}\) where \(\text{T} =\) tails and \(\text{H} =\) heads. The sample space has four outcomes. \(\text{A} =\) getting one head. There are two outcomes that meet this condition \(\text{\{HT, TH\}}\), so \(P(\text{A}) = \frac{2}{4} = 0.5\).
Suppose you roll one fair six-sided die, with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event \(\text{E} =\) rolling a number that is at least five. There are two outcomes {5, 6}. \(P(\text{E}) = \frac{2}{6}\). If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, \(\frac{2}{6}\) of the rolls would result in an outcome of "at least five". You would not expect exactly \(\frac{2}{6}\). The long-term relative frequency of obtaining this result would approach the theoretical probability of \(\frac{2}{6}\) as the number of repetitions grows larger and larger.
Definition: Law of Large Numbers
This important characteristic of probability experiments is known as the law of large numbers which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word empirical is often used instead of the word observed.)
It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be unfair , or biased . Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased.  Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.
The "OR" Event
An outcome is in the event \(\text{A OR B}\) if the outcome is in \(\text{A}\) or is in \(\text{B}\) or is in both \(\text{A}\) and \(\text{B}\). For example, let \(\text{A} = \{1, 2, 3, 4, 5\}\) and \(\text{B} = \{4, 5, 6, 7, 8\}\). \(\text{A OR B} = \{1, 2, 3, 4, 5, 6, 7, 8\}\). Notice that 4 and 5 are NOT listed twice.
The "AND" Event
An outcome is in the event \(\text{A AND B}\) if the outcome is in both \(\text{A}\) and \(\text{B}\) at the same time. For example, let \(\text{A}\) and \(\text{B}\) be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then \(\text{A AND B} = {4, 5}\).
The complement of event \(\text{A}\) is denoted \(\text{A'}\) (read "A prime"). \(\text{A'}\) consists of all outcomes that are NOT in \(\text{A}\). Notice that
\[P(\text{A}) + P(\text{A′}) = 1. \nonumber\]
For example, let \(\text{S} = \{1, 2, 3, 4, 5, 6\}\) and let \(\text{A} = {1, 2, 3, 4}\). Then, \(\text{A′} = {5, 6}\) and \(P(A) = \frac{4}{6}\), \(P(\text{A′}) = \frac{2}{6}\), and
\[P(\text{A}) + P(\text{A′}) = \frac{4}{6} + \frac{2}{6} = 1. \nonumber\]
The conditional probability of \(\text{A}\) given \(\text{B}\) is written \(P(\text{A|B})\). \(P(\text{A|B})\) is the probability that event \(\text{A}\) will occur given that the event \(\text{B}\) has already occurred. A conditional reduces the sample space . We calculate the probability of \(\text{A}\) from the reduced sample space \(\text{B}\). The formula to calculate \(P(\text{A|B})\) is
\[P(\text{A|B}) = \frac{\text{P(A AND B)}}{\text{P(B)}} \nonumber\]
where \(P(\text{B})\) is greater than zero.
For example, suppose we toss one fair, six-sided die. The sample space \(\text{S} = \{1, 2, 3, 4, 5, 6\}\).  To calculate \(P(\text{A|B})\), we count the number of outcomes 2 or 3 in the sample space \(\text{B} = \{2, 4, 6\}\). Then we divide that by the number of outcomes \(\text{B}\) (rather than \(\text{S}\)).
We get the same result by using the formula. Remember that \(\text{S}\) has six outcomes.
\[ \begin{align*} P(\text{A|B}) &= \dfrac{ \text{ P(A AND B) } } {P(\text{B})} \\[4pt] &= \dfrac{\dfrac{\text{the number of outcomes that are 2 or 3 and even in S}}{6}}{\dfrac{\text{the number of outcomes that are even in S}}{6}} \\[4pt] &= \dfrac{\frac{1}{6}}{\frac{3}{6}} = \dfrac{1}{3} \end{align*}\]
Understanding Terminology and Symbols
It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.
Example \(\PageIndex{1}\)
The sample space \(S\) is the whole numbers starting at one and less than 20.
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\(S =\) _____________________________
Let event \(A =\) the even numbers and event \(B =\) numbers greater than 13.
- \(A =\) _____________________, \(B =\) _____________________
- \(P(\text{A}) =\) _____________, \(P(\text{B}) =\) ________________
- \(\text{A AND B} =\) ____________________, \(\text{A OR B} =\) ________________
- \(P(\text{A AND B}) =\) _________, \(P(\text{A OR B}) =\) _____________
- \(\text{A′} =\) _____________, \(P(\text{A′}) =\) _____________
- \(P(\text{A}) + P(\text{A′}) =\) ____________
- \(P(\text{A|B}) =\) ___________, \(P(\text{B|A}) =\) _____________; are the probabilities equal?
Answer
- \(\text{S} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19\}\)
- \(\text{A} = \{2, 4, 6, 8, 10, 12, 14, 16, 18\}, \text{B} = \{14, 15, 16, 17, 18, 19\}\)
- \(P(\text{A}) = \frac{9}{19}\), \(P(\text{B}) = \frac{6}{19}\)
- \(\text{A AND B} = \{14,16,18\}\), \(\text{A OR B} = \{2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19\}\)
- \(P(\text{A AND B}) = \frac{3}{19}\), \(P(\text{A OR B}) = \frac{12}{19}\)
- \(\text{A′} = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19\); \(P(\text{A′}) = \frac{10}{19}\)
- \(P(\text{A}) + P(\text{A′}) = 1\left((\frac{9}{19} + \frac{10}{19} = 1\right)\)
- \(P(\text{A|B}) = \frac{\text{P(A AND B)}}{\text{P(B)}} = \frac{3}{6}, P(\text{B|A}) = \frac{\text{P(A AND B)}}{\text{P(A)}} = \frac{3}{9}\), No
Exercise \(\PageIndex{1}\)
The sample space S is the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: (1, 4)).
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\(S =\) _____________________________
Let event \(A =\) the sum is even and event \(B =\) the first number is prime. - \(A =\) _____________________, \(B =\) _____________________
- \(P(\text{A}) =\) _____________, \(P(\text{B}) =\) ________________
- \(\text{A AND B} =\) ____________________, \(\text{A OR B} =\) ________________
- \(P(\text{A AND B}) =\) _________, \(P(\text{A OR B}) =\) _____________
- \(\text{B′} =\) _____________, \(P(\text{B′)} =\) _____________
- \(P(\text{A}) + P(\text{A′}) =\) ____________
- \(P(\text{A|B}) =\) ___________, \(P(\text{B|A}) =\) _____________; are the probabilities equal?
Answer
- \(\text{S} = \{(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)\}\)
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\(\text{A} = \{(1,1), (1,3), (2,2), (2,4), (3,1), (3,3)\}\)
\(\text{B} = \{(2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)\}\) - \(P(\text{A}) = \frac{1}{2}\), \(P(\text{B}) = \frac{2}{3}\)
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\(\text{A AND B} = \{(2,2), (2,4), (3,1), (3,3)\}\)
\(\text{A OR B} = \{(1,1), (1,3), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)\}\) - \(P(\text{A AND B}) = \frac{1}{3}, P(\text{A OR B}) = \frac{5}{6}\)
- \(\text{B′} = \{(1,1), (1,2), (1,3), (1,4)\}, P(\text{B′}) = \frac{1}{3}\)
- \(P(\text{B}) + P(\text{B′}) = 1\)
- \(P(\text{A|B}) = \frac{P(\text{A AND B})}{P(\text{B})} = \frac{1}{2}, P(\text{B|A}) = \frac{P(\text{A AND B})}{P(\text{B})} = \frac{2}{3}\), No.
Example \(\PageIndex{2A}\)
A fair, six-sided die is rolled. Describe the sample space S , identify each of the following events with a subset of S and compute its probability (an outcome is the number of dots that show up).
- Event \(\text{T} =\) the outcome is two.
- Event \(\text{A} =\) the outcome is an even number.
- Event \(\text{B} =\) the outcome is less than four.
- The complement of \(\text{A}\).
- \(\text{A GIVEN B}\)
- \(\text{B GIVEN A}\)
- \(\text{A AND B}\)
- \(\text{A OR B}\)
- \(\text{A OR B′}\)
- Event \(\text{N} =\) the outcome is a prime number.
- Event \(\text{I} =\) the outcome is seven.
Solution
- \(\text{T} = \{2\}\), \(P(\text{T}) = \frac{1}{6}\)
- \(A = \{2, 4, 6\}\), \(P(\text{A}) = \frac{1}{2}\)
- \(\text{B} = \{1, 2, 3\}\), \(P(\text{B}) = \frac{1}{2}\)
- \(\text{A′} = \{1, 3, 5\}, P(\text{A′}) = \frac{1}{2}\)
- \(\text{A|B} = \{2\}\), \(P(\text{A|B}) = \frac{1}{3}\)
- \(\text{B|A} = \{2\}\), \(P(\text{B|A}) = \frac{1}{3}\)
- \(\text{A AND B} = {2}, P(\text{A AND B}) = \frac{1}{6}\)
- \(\text{A OR B} = \{1, 2, 3, 4, 6\}\), \(P(\text{A OR B}) = \frac{5}{6}\)
- \(\text{A OR B′} = \{2, 4, 5, 6\}\), \(P(\text{A OR B′}) = \frac{2}{3}\)
- \(\text{N} = \{2, 3, 5\}\), \(P(\text{N}) = \frac{1}{2}\)
- A six-sided die does not have seven dots. \(P(7) = 0\).
Example \(\PageIndex{2B}\)
Table describes the distribution of a random sample \(S\) of 100 individuals, organized by gender and whether they are right- or left-handed.
| Right-handed | Left-handed | |
|---|---|---|
| Males | 43 | 9 |
| Females | 44 | 4 |
Let’s denote the events \(M =\) the subject is male, \(F =\) the subject is female, \(R =\) the subject is right-handed, \(L =\) the subject is left-handed. Compute the following probabilities:
- \(P(\text{M})\)
- \(P(\text{F})\)
- \(P(\text{R})\)
- \(P(\text{L})\)
- \(P(\text{M AND R})\)
- \(P(\text{F AND L})\)
- \(P(\text{M OR F})\)
- \(P(\text{M OR R})\)
- \(P(\text{F OR L})\)
- \(P(\text{M'})\)
- \(P(\text{R|M})\)
- \(P(\text{F|L})\)
- \(P(\text{L|F})\)
Answer
- \(P(\text{M}) = 0.52\)
- \(P(\text{F}) = 0.48\)
- \(P(\text{R}) = 0.87\)
- \(P(\text{L}) = 0.13\)
- \(P(\text{M AND R}) = 0.43\)
- \(P(\text{F AND L}) = 0.04\)
- \(P(\text{M OR F}) = 1\)
- \(P(\text{M OR R}) = 0.96\)
- \(P(\text{F OR L}) = 0.57\)
- \(P(\text{M'}) = 0.48\)
- \(P(\text{R|M}) = 0.8269\) (rounded to four decimal places)
- \(P(\text{F|L}) = 0.3077\) (rounded to four decimal places)
- \(P(\text{L|F}) = 0.0833\)
References
- “Countries List by Continent.” Worldatlas, 2013. Available online at http://www.worldatlas.com/cntycont.htm (accessed May 2, 2013).
Review
In this module we learned the basic terminology of probability. The set of all possible outcomes of an experiment is called the sample space. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.
Formula Review
\(\text{A}\) and \(\text{B}\) are events
\(P(\text{S}) = 1\) where \(\text{S}\) is the sample space
\(0 \leq P(\text{A}) \leq 1\)
\(P(\text{A|B}) = \frac{\text{P(A AND B)}}{\text{P(B)}}\)
Glossary
- Conditional Probability
- the likelihood that an event will occur given that another event has already occurred
- Equally Likely
- Each outcome of an experiment has the same probability.
- Event
- a subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a sample space and is usually denoted by \(S\). An event is an arbitrary subset in \(S\). It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital letters such as \(A, B, C\), and so on.
- Experiment
- a planned activity carried out under controlled conditions
- Outcome
- a particular result of an experiment
- Probability
- a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; the foundation of statistics is given by the following 3 axioms (by A.N. Kolmogorov, 1930’s): Let \(S\) denote the sample space and \(A\) and \(B\) are two events in S . Then:
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- \(0 \leq P(\text{A}) \leq 1\)
- If \(\text{A}\) and \(\text{B}\) are any two mutually exclusive events, then \(\text{P}(\text{A OR B}) = P(\text{A}) + P(\text{B})\).
- \(P(\text{S}) = 1\)
- Sample Space
- the set of all possible outcomes of an experiment
- The AND Event
- An outcome is in the event \(\text{A AND B}\) if the outcome is in both \(\text{A AND B}\) at the same time.
- The Complement Event
- The complement of event \(\text{A}\) consists of all outcomes that are NOT in \(\text{A}\).
- The Conditional Probability of A GIVEN B
- \(P(\text{A|B})\) is the probability that event \(\text{A}\) will occur given that the event \(\text{B}\) has already occurred.
- The Or Event
- An outcome is in the event \(\text{A OR B}\) if the outcome is in \(\text{A}\) or is in \(\text{B}\) or is in both \(\text{A}\) and \(\text{B}\).
Exercise 3.2.2
In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the symbols for the probabilities of the events for parts a through j. (Note that you cannot find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)
- Let \(\text{F}\) be the event that a student is female.
- Let \(\text{M}\) be the event that a student is male.
- Let \(\text{S}\) be the event that a student has short hair.
- Let \(\text{L}\) be the event that a student has long hair.
- The probability that a student does not have long hair.
- The probability that a student is male or has short hair.
- The probability that a student is a female and has long hair.
- The probability that a student is male, given that the student has long hair.
- The probability that a student has long hair, given that the student is male.
- Of all the female students, the probability that a student has short hair.
- Of all students with long hair, the probability that a student is female.
- The probability that a student is female or has long hair.
- The probability that a randomly selected student is a male student with short hair.
- The probability that a student is female.
Answer
- \(P(\text{L′)} = P(\text{S})\)
- \(P(\text{M OR S})\)
- \(P(\text{F AND L})\)
- \(P(\text{M|L})\)
- \(P(\text{L|M})\)
- \(P(\text{S|F})\)
- \(P(\text{F|L})\)
- \(P(\text{F OR L})\)
- \(P(\text{M AND S})\)
- \(P(\text{F})\)
Use the following information to answer the next four exercises.  It contains 12 hats, 15 noisemakers, ten finger traps, and five bags of confetti.
Let \(H =\) the event of getting a hat.
Let \(N =\) the event of getting a noisemaker.
Let \(F =\) the event of getting a finger trap.
Let \(C =\) the event of getting a bag of confetti.
Exercise 3.2.3
Find \(P(\text{H})\).
Exercise 3.2.4
Find \(P(\text{N})\).
Answer
\(P(\text{N}) = \frac{15}{42} = \frac{5}{14} = 0.36\)
Exercise 3.2.5
Find \(P(\text{F})\).
Exercise 3.2.6
Find \(P(\text{C})\).
Answer
\(P(\text{C}) = \frac{5}{42} = 0.12\)
Use the following information to answer the next six exercises. A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange.
Let \(B =\) the event of getting a blue jelly bean
Let \(G =\) the event of getting a green jelly bean.
Let \(O =\) the event of getting an orange jelly bean.
Let \(P =\) the event of getting a purple jelly bean.
Let \(R =\) the event of getting a red jelly bean.
Let \(Y =\) the event of getting a yellow jelly bean.
Exercise 3.2.7
Find \(P(\text{B})\).
Exercise 3.2.8
Find \(P(\text{G})\).
Answer
\(P(\text{G}) = \frac{20}{150} = \frac{2}{15} = 0.13\)
Exercise 3.2.9
Find \(P(\text{P})\).
Exercise 3.2.10
Find \(P(\text{R})\).
Answer
\(P(\text{R}) = \frac{22}{150} = \frac{11}{75} = 0.15\)
Exercise 3.2.11
Find \(P(\text{Y})\).
Exercise 3.2.12
Find \(P(\text{O})\).
Answer
\(P(text{O}) = \frac{150-22-38-20-28-26}{150} = \frac{16}{150} = \frac{8}{75} = 0.11\)
Use the following information to answer the next six exercises. There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region).
Let \(\text{A} =\) the event that a country is in Asia.
Let \(\text{E} =\) the event that a country is in Europe.
Let \(\text{F} =\) the event that a country is in Africa.
Let \(\text{N} =\) the event that a country is in North America.
Let \(\text{O} =\) the event that a country is in Oceania.
Let \(\text{S} =\) the event that a country is in South America.
Exercise 3.2.13
Find \(P(\text{A})\).
Exercise 3.2.14
Find \(P(\text{E})\).
Answer
\(P(\text{E}) = \frac{47}{194} = 0.24\)
Exercise 3.2.15
Find \(P(\text{F})\).
Exercise 3.2.16
Find \(P(\text{N})\).
Answer
\(P(\text{N}) = \frac{23}{194} = 0.12\)
Exercise 3.2.17
Find \(P(\text{O})\).
Exercise 3.2.18
Find \(P(\text{S})\).
Answer
\(P(\text{S}) = \frac{12}{194} = \frac{6}{97} = 0.06\)
Exercise 3.2.19
What is the probability of drawing a red card in a standard deck of 52 cards?
Exercise 3.2.20
What is the probability of drawing a club in a standard deck of 52 cards?
Answer
\(\frac{13}{52} = \frac{1}{4} = 0.25\)
Exercise 3.2.21
What is the probability of rolling an even number of dots with a fair, six-sided die numbered one through six?
Exercise 3.2.22
What is the probability of rolling a prime number of dots with a fair, six-sided die numbered one through six?
Answer
\(\frac{3}{6} = \frac{1}{2} = 0.5\)
Use the following information to answer the next two exercises. You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color wheel is equal in area.
Let \(\text{B} =\) the event of landing on blue.
Let \(\text{R} =\) the event of landing on red.
Let \(\text{G} =\) the event of landing on green.
Let \(\text{Y} =\) the event of landing on yellow.
Exercise 3.2.23
If you land on \(\text{Y}\), you get the biggest prize. Find \(P(\text{Y})\).
 What is \(P(\text{R})\)?
Answer
\(\text{P}(R) = \frac{4}{8} = 0.5\)
Use the following information to answer the next ten exercises. On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters.
Let \(\text{I} =\) the event that a player in an infielder.
Let \(\text{O} =\) the event that a player is an outfielder.
Let \(\text{H} =\) the event that a player is a great hitter.
Let \(\text{N} =\) the event that a player is not a great hitter.
Exercise 3.2.25
Write the symbols for the probability that a player is not an outfielder.
Exercise 3.2.26
Write the symbols for the probability that a player is an outfielder or is a great hitter.
Answer
\(P(\text{O OR H})\)
Exercise 3.2.27
Write the symbols for the probability that a player is an infielder and is not a great hitter.
Exercise 3.2.28
Write the symbols for the probability that a player is a great hitter, given that the player is an infielder.
Answer
\(P(\text{H|I})\)
Exercise 3.2.29
Write the symbols for the probability that a player is an infielder, given that the player is a great hitter.
Exercise 3.2.30
Write the symbols for the probability that of all the outfielders, a player is not a great hitter.
Answer
\(P(\text{N|O})\)
Exercise 3.2.31
Write the symbols for the probability that of all the great hitters, a player is an outfielder.
Exercise 3.2.32
Write the symbols for the probability that a player is an infielder or is not a great hitter.
Answer
\(P(\text{I OR N})\)
Exercise 3.2.33
Write the symbols for the probability that a player is an outfielder and is a great hitter.
Exercise 3.2.34
Write the symbols for the probability that a player is an infielder.
Answer
\(P(\text{I})\)
Exercise 3.2.35
What is the word for the set of all possible outcomes?
Exercise 3.2.36
What is conditional probability?
Answer
The likelihood that an event will occur given that another event has already occurred.
Exercise 3.2.37
A shelf holds 12 books. Eight are fiction and the rest are nonfiction. Each is a different book with a unique title. The fiction books are numbered one to eight. The nonfiction books are numbered one to four. Randomly select one book
Let \(\text{F} =\) event that book is fiction
Let \(\text{N} =\) event that book is nonfiction
What is the sample space?
Exercise 3.2.38
What is the sum of the probabilities of an event and its complement?
Answer
1
Use the following information to answer the next two exercises. You are rolling a fair, six-sided number cube. Let \(\text{E} =\) the event that it lands on an even number. Let \(\text{M} =\) the event that it lands on a multiple of three.
Exercise 3.2.39
What does \(P(\text{E|M})\) mean in words?
Exercise 3.2.40
What does \(P(\text{E OR M})\) mean in words?
Answer
the probability of landing on an even number or a multiple of three


12.4: Discussion- Oroonoko
Source reading : Behn, Oroonoko
Discussion Prompt
Answer any three questions.
- In what sense could Oroonoko be said to be heroic? What are his heroic characteristics? What other literary heroes would you compare him to? Explain.
- How does this novel portray slavery? What are Oroonoko’s attitudes toward slavery? What are the narrator’s attitudes?
- How does the story portray the love between Oroonoko and Imoinda? Why can’t their love survive in Surinam?
- Why does Oroonoko kill Imoinda? In what sense can his action be said to be justified? In what ways can it never be justified?
- Consider the way that violence can be seen to be just beneath the surface of civilization in this story, especially concerning the execution of Oroonoko? What role does this scene play in the story? Is it appropriate? Why or why not?
Discussion Board Instructions
Answer each question, unless otherwise specified. Number your answers (do not copy the prompt into your answers.) Answers need not be of equal length, but all answers must be well developed. The word count total (250+) refers to all answers added together.
In the first sentence or two of your first answer identify the name of the work and full name of the author (if applicable).
Include at least one quotation from the work in your overall response. Use line number(s) to identify the location of your citation. No works cited page is needed.
Grading Rubric
| Levels of Achievement | |||||
|---|---|---|---|---|---|
| Criteria | Excellent | Good | Adequate | Inadequate | Unacceptable |
| Response: Content Length and Development | 22 Points | 19 Points | 16 Points | 11 Points | 0 Points |
| Meets or exceeds the minimum length requirement. Demonstrates insight and understanding of the text(s) in an excellent, thorough, detailed, clear, and well developed response to the entire prompt. Response is comprised primarily of the student’s own answer(s), but is supported by at least one well chosen quotation. | Meets the minimum length requirement. Demonstrates understanding of the text(s) in a good, clear, well developed response to the entire prompt. Response is comprised primarily of the student’s own answer(s), but is supported by at least one well chosen quotation. | Meets or nearly meets the minimum length requirement. Demonstrates a basic understanding of the text(s) in a developed response to the prompt. Includes an adequate amount of the student’s own answer(s), but may incorporate too much summary, paraphrase, or quotations or lack a quotation for support. | Meets or falls 15%+ short of the minimum length requirement. Inadequately responds to the prompt with answer(s) that are substantively incorrect, superficial, underdeveloped, and/or reliant mostly on summary, paraphrase, or quotations. | Unacceptable development: Fails to meet at least 50% of the length requirement. | |
| Response: Source Identification and Citation | 4 points | 3.5 points | 3 Points | 2 Points | 0 Points |
| Clearly identifies by title and full name of author the name(s) of work(s) to be discussed. Cites material from source(s) as instructed in the assignment. | Identifies the work(s) and author(s). Cites material from source(s) as instructed in the assignment. | Identifies the work(s) or author(s). Cites adequately. | Inadequately identifies source(s) and author(s) and/or inadequately cites source(s). | Lacks identification and citation of source(s). | |
| Response: Grammar and Mechanics | 4 Points | 3.5 Points | 3 Points | 2 Points | 0 Points |
| Response is free or nearly free of errors in grammar (including but not limited to use of full sentences and appropriate word choice) and mechanics (including but not limited to spelling and punctuation). | Response demonstrates occasional errors in grammar or mechanics that do not affect overall clarity. | Demonstrates errors in grammar or mechanics that occasionally affect meaning. | Inadequately meets standards for grammar and mechanics. Clarity is frequently affected. | Unacceptable level of errors in grammar or mechanics. Clarity is greatly affected. | |
| Comment 1: Engagement and Content | 7 Points | 6 Points | 5 Points | 3.5 Points | 0 Points |
| Demonstrates thorough engagement in a comment that is detailed, well crafted, and reflective. | Demonstrates effective engagement in a comment that is effectively crafted and reflective. | Demonstrates an adequate engagement in a comment that addresses at least one point of an answer to the other student’s response. | Demonstrates inadequate engagement in a comment that only superficially or tangentially addresses the content of the other student’s response. | Unacceptable development: Fails to meet at least 50% of the minimum length requirement. | |
| Comment 1: Grammar and Mechanics and Length Requirement | 3 Points | 2.5 Points | 2 Points | 1.5 Points | 0 Points |
| Contains few if any grammar or mechanics issues. Meets or exceeds the minimum length requirement. | Contains few if any grammar or mechanics issues. Meets the length requirement. | Contains grammar and/or mechanics issues that occasionally affect clarity. Meets or falls within 10% of the minimum length requirement. | Inadequately meets standards for grammar and mechanics. Clarity is frequently affected. Meets or falls 15%+ short of the minimum length requirement. | Unacceptable level of errors in grammar or mechanics that greatly affect clarity. (No credit for this element if fails to meet at least 50% of the minimum length requirement.) | |
| Comment 2: Engagement and Content | 7 Points | 6 Points | 5 Points | 3.5 Points | 0 Points |
| Demonstrates thorough engagement with the content of the other student’s response in a comment that is detailed, well crafted, and reflective. | Demonstrates effective engagement in a comment that is effectively crafted and reflective. | Demonstrates an adequate engagement in a comment that engages with at least one point of content in the other student’s response. | Demonstrates inadequate engagement in a comment that only superficially or tangentially addresses the content of the other student’s response. | Unacceptable development: Fails to meet at least 50% of the length requirement. | |
| Comment 2: Grammar and Mechanics and Length Requirement. | 3 Points | 2.5 Points | 2 Points | 1.5 Points | 0 Points |
| Contains few if any grammar or mechanics issues. Meets or exceeds the minimum length requirement. | Contains a few grammar and/or mechanics issues, but these do not affect overall clarity. Meets the minimum length requirement. | Contains grammar and/or mechanics issues that occasionally affect clarity. Meets or falls within 10% of the minimum length requirement. | Inadequately meets standards for grammar and mechanics. Clarity is frequently affected. Meets or falls 15%+ short of the minimum length requirement. | Unacceptable level of errors in grammar or mechanics that greatly affect clarity. (No credit for this element if comment fails to meet at least 50% of the minimum length requirement.) |
- Survey of English Literature I. Authored by : Wendy Howard Gray. Provided by : Reynolds Community College. Located at : http://www.reynolds.edu/ . License : CC BY: Attribution


16.1: The Community as the Client
By the end of this section, you should be able to:
- Explain how the community can serve as the focus of care.
- Describe a community by location, population, and social systems.
- Develop a community profile.
Broadly, a community is defined as a group of people with at least one characteristic in common. The characteristic may be a place, a personal attribute, or a common goal. In each community, there are individuals with health problems and health risks, some of which may be individual, meaning the rest of the community does not face similar circumstances. For example, a person who smokes or lives in a home with smokers faces increased respiratory health risks compared to a person who does not smoke or live with smokers. A person who does not wear sunscreen has an increased risk of skin cancer compared to a person who applies sunscreen daily. However, every community may house a common health condition or shared health risk . For example, a community situated near a factory that disposes of its toxic chemicals in the water supply shares the risks of consuming contaminated drinking water. Similarly, residents of a city that provides automated external defibrillators (AEDs) in high-traffic public places may have a better chance of surviving a cardiac event than those with no accessible AEDs.
In population- or community-based nursing, nurses work to address a community’s collective health instead of individual health needs. Recall the chapter’s opening scenario. Try to list five communities that Elsa belongs to.
Community Attributes
Formal and informal communities exist within and across towns and cities. A community may comprise people who celebrate the same cultural holidays or people who are at risk for the same health condition.
Consider the many communities that may be present in a classroom of nursing students: a community of students, a clinical team, a study group, students with children, and perhaps students studying nursing as a new career. Elsa, from the chapter’s opening scenario, is a member of numerous communities, including residents of the town, people with type 1 diabetes mellitus, high school students who are gender nonconforming , the soccer team, and residents who get around on bikes, among others. Communities of nursing clients may consist of those who inhabit a specific geographic location, share specific characteristics, or access the same social systems. Communities may also form when groups cannot access select health services and social systems.
Individuals do not have to live within mapped boundaries to be part of a geographic community. People who work, worship, or play in an area where they do not live can be interested in that community’s health.
A key activity for geographic communities is civic engagement —involvement in issues of public concern through political and nonpolitical processes that increase community vitality, challenge injustices, and address social problems (Figure 16.2) (American Psychological Association, 2009; Wray-Lake & Abrams, 2020). Other geographically based communities include members of community centers, such as a Council on Aging or a Boys and Girls Club, and populations of school students, home caregivers, and teachers in a geographic community.
Social determinants of health and civic engagement are closely linked. Healthy People 2030 identifies voting, volunteering, participating in group activities and activities such as community gardening, and participating in a recreational sports team as methods of civic participation. Such activities can benefit the health of individuals and populations. Healthy People 2030 aims to increase the proportion of voting-age citizens who vote and increase knowledge of civic activities or interventions that may ameliorate health disparities.
People who share personal characteristics such as race , ethnicity , gender , age, or occupation may create a formal community for mutual support or to advance equitable practices across health care and other fields. Others who share such characteristics may not participate in a formal community but may informally associate with members of the community, remain knowledgeable of the shared successes and challenges the community faces, and advocate for change. Populations of individuals who share health risks may also constitute a community.
Social Media and Other Social Systems
Social media and other social systems provide avenues for the formation and definition of communities. Social and political networks may serve as a basis for uniting as a community for health change. Technology has dramatically enhanced opportunities to join a community, as many people use social networks to participate in civic engagement, learn about health, effect change through sharing stories and knowledge, and organize to advocate for policies and societal changes that advance health. Social media has increased the visibility of community health issues, as community members and advocates can quickly share information with people around the globe.
Advocacy in the context of community health refers to support for a specific health cause or recommendations for community and policy change . Advocacy is a significant health promotion strategy to inform voters and policymakers of pressing community health issues and the consequences of harmful policies or inaction (Jackson et al., 2021). Community members who share their own health experiences encourage the forming of community, as other members who identify with the experiences or want to support health change can join the conversation and become involved. Social media has provided a platform for community members, clinicians, organizations, charities, and others to connect and work together to improve the dimensions of community health in many areas. Health advocates have used social media to call attention to, and impact change for, many issues across the globe, such as alcohol use in the Netherlands, tobacco control in Australia, sexual harassment in Egypt, traffic accidents in Ireland, maternal mortality in the United States, and housing in New Zealand (Albalawi & Sixsmith, 2015; Behm et al., 2022; Bekkers et al., 2011; Chisholm & O’Sullivan, 2017; Hefler et al., 2013; Peuchaud, 2014). Advocating for Population Health covers nurse advocacy in more detail.
Communities have worked to generate health change through social media messaging and campaigns. Over the last few years, many social media movements have raised awareness of select health issues, changed public perception of health problems, and even successfully advocated for policy change in companies and organizations or for greater society through changes to public law. For example, the #MeToo movement that went viral in 2017 contributed to the U.S. Congress passing of the Sexual Assault Survivors’ Bill of Rights.
Using the social media platform of your choice, explore one of the hashtags below, and then respond to the questions that follow.
- #BlackBreastfeedingWeek
- #TransHealthMatters
- #ReclaimingMySize
- #NativeHealthMatters
- #NothingAboutUsWithoutUs
- #MentalHealthAwareness
- #NotJustSkinDeep
- #WomensHealthMatters
- #EveryBodyHasAStory
- #MedicareForAll
- #BlackLivesMatter
- #ThisIsEndometriosis
- #GirlsCount
- #HIVNotRetro
- #FarmworkersFeedUs
- #TransLivesMatter
- #HealthEquityNow
- #QueerAndDisabled
- #HealthForAll
- #BreakTheStigma
- #BlackHealthMatters
- #SaludLatinx
- What is the health message or health change that users of the hashtag hope to make?
- What are the characteristics of community members amplifying the health message? Are any voices being amplified that would be silenced without social media?
- Since the implementation of the hashtag, has any health change occurred? Think about health change locally, statewide, and nationally.
Community Assessments and Profiles
Assessment is the first and most important step of the nursing process. Nurses cannot address a problem if they do not know it exists, and they cannot support a health behavior or community effort if they do not realize it is occurring. Assessment begins with gathering data. Nurses may take several approaches to gathering data. First, they may conduct a windshield survey . In a windshield survey, nurses explore a community or neighborhood and observe on foot, from a bench or street corner, or through the windows of a vehicle (Centers for Disease Control and Prevention, 2010). Observations made using this type of survey can alert nurses to urgent or potential health risks in the community, as observers may notice the condition of sidewalks and public spaces, community noise levels, street access and activity, and the presence or absence of businesses and places of employment. See Appendix A for an example of a windshield survey template that nurses may use to conduct one of these assessments. Per the University of Kansas Center for Community Health and Development Community Tool Box (2023), during such a survey, nurses should observe:
- Housing : age and condition of homes in the neighborhood
- Other buildings: occupancy and accessibility of buildings
- Public spaces: availability of places to gather and their amenities, upkeep, and utilization
- Parks: utilization, times of use, activities facilitated
- Culture and entertainment: presence of museums, libraries, theaters, and sports venues and their accessibility and use by members representative of the full community
- Streetscape: condition of building facades and storefronts, presence of trees and sidewalks, cleanliness
- Street use: presence of people during various times of the day
- Commercial activity: types of businesses, storefront vacancies, stores that provide necessities like groceries
- Signs: languages, level of information
- Industry: types of industry, pollution
- Land use: open space compared to used land; distribution of residential, commercial, and industrial areas
- Infrastructure: condition of roads, bridges, and sidewalks; differences in condition across the community
- Public transportation : presence and use of system
- Traffic: level of traffic, vehicles in traffic
- Environmental quality: green space presence and distribution, air quality, water sources
- Race/ethnicity: characteristics of community, separation or cohesion of groups
- Faith communities : religious institution types and presence
- Health services: hospital and clinic presence and accessibility
- Community and public services: presence of mental health centers, food banks, and housing support and their distribution and accessibility by public transit
- Community safety: police and fire stations
- Public schools: presence of schools and their level of maintenance or upkeep
- Higher education: two- and four-year colleges, location, integration with community versus isolation
- Political activity: political signs (e.g., yard signs during elections), protests, demonstrations
- Community organizations: service clubs and other groups centered on community issues
- Media: local outlets, sponsorship/influence
- Differences among neighborhoods: states of maintenance, presence of community amenities across neighborhoods or sections
- The “feel” of the community: the observer’s overall impression of the community
Regardless of the method they use to conduct a windshield survey, nurses should conduct the survey in pairs or groups. First, it is unsafe to attempt to drive and note observations at the same time. Additionally, each nurse may have different observations, which can add to the richness of data collected. Nurses may also collect data from open sources in the community, such as publicly available reports, census data, and other public documents (Quinn et al., 2019). For example, department of health reports regarding the rates of communicable illnesses and chronic conditions can help elucidate a community’s health needs, and the posted minutes of a local select board meeting can illuminate health topics on the minds of local community partners and residents. See Assessment, Analysis, and Diagnosis for more information regarding community data sources. Finally, nurses should consult key community partners, policymakers, community leaders, and community members during a community assessment.
These two videos were created by nursing students as they conducted their own windshield surveys. The first video displays a windshield survey of a community in Ohio with the students’ interpretation of their observations. The second video facilitates a virtual windshield survey of communities in Virginia.
Watch both videos, and then respond to the following questions.
- How do you interpret the windshield survey of communities in Virginia?
- What other observations would you make of the Virginia communities to prepare a community assessment like the one displayed for the Ohio community?
- What conclusions might you draw about health and well-being in the Virginia communities based on the windshield survey?
Including community members in community-based nursing work is essential and should start from the earliest stages of a community initiative. Involving community members’ perspectives and participation supports not only thorough assessments and meaningful change but also ethical practice. Meaningful partnership s with community members show respect for the community, build trust and relationships, minimize risk and burden, build legitimacy, and positively impact health (Adhikari et al., 2020; Haldane et al., 2019).
Community inclusion also helps nurses work against both the narrative and the reality that clinicians often tell people what to do without considering personal, contextual, social, or other factors. Such actions may be perceived as patronizing and marginalizing the community, or treating them as insignificant and making them feel powerless or unimportant. In nursing, decolonization refers to processes that systematically dismantle “western colonial ideologies of superiority, thought, approaches, and privilege ” (Fontenot & McMurray, 2020, p. 272). Partnering with members of a community is in alignment with decolonization and can contribute to community interventions that are robust, respectful, culturally responsive, and inclusive (Lin et al., 2020; McCreedy et al., 2018).
Colonialism is an ideology with a strong foundation of control and influence. Nursing students may remember learning about colonists in the contexts of history, war, and geography during primary and secondary school. Depending on the country in which a nursing student attended primary and secondary school, colonists and colonialism may have been depicted as a positive story of starting a new life. However, colonists invade, conquer, occupy, dominate, and exploit the lands of other people. Colonialism and racism, the unfair treatment of individuals based on race, share these harmful foundations. Colonialism and racism are both present in nursing; power dynamics, forced dependency, lack of self-determination, and othering shape nursing education, research, and practice as well as health policy. In community health, nurses must ensure meaningful community participation in community assessments and population health program planning in order to confront racism and colonialism in the discipline.
(See Fontenot & McMurray, 2020; Kimani, 2023.)
Gordon’s functional health patterns provide a method of structuring a community assessment and organizing the resulting data (Gordon, 1982). Nurse theorist Marjory Gordon developed the functional health patterns to guide nurses through a thorough assessment of individual clients and families (Gordon, 1982). However, the functional health patterns can also be applied to a community assessment. Ten years after Gordon published her work, nurse scholars Nancy Kriegler and Marilyn Harton used it to develop the community health assessment tool (Kriegler & Harton, 1992). See Table 16.1 for descriptions of the 11 functional health patterns for use in community assessments and the items nurses may assess in each pattern.
| Pattern | Community Application | Assessment Items |
|---|---|---|
| Health perception–health management | Community provision of health and safety |
|
| Nutritional-metabolic | Community provision of nutritional needs, including education |
|
| Elimination | Community handling of waste |
|
| Activity-exercise | Community provision of transportation and recreational opportunities |
|
| Cognitive-perceptual | Community decision-making |
|
| Sleep-rest | Community experience and expression of rhythms and cycles |
|
| Self-perception–self-concept | Community view of itself |
|
| Role-relationship | Community definition of its own role through internal informal and formal relationships and interactions with other communities |
|
| Sexuality-reproductive | Community provision of resources and education for reproductive health and family structures |
|
| Coping–stress tolerance | Community provision of support services |
|
| Value-belief | Community addressment of spiritual, cultural, and ethical needs |
|
A functional health pattern assessment can help the nurse to develop a community profile that includes identifying health problems, describing health risks, and determining health promotion activities. Recall the chapter’s opening scenario. What patterns does the description of Elsa’s community address? What questions would the community health nurse have for Elsa and other community members to develop a comprehensive community assessment?
The functional health pattern assessment applies to nursing assessment and practice and can be used during interprofessional community health endeavors. Other tools that may be used include the American Hospital Association’s Community Health Assessment Toolkit , the CDC’s action model, and the CDC’s Community Health Assessment and Group Evaluation (CHANGE) tool. Assessment, Analysis, and Diagnosis provides more information on community health assessment tools, models, and frameworks.
Read the scenario, and then respond to the questions that follow.
Think back to the chapter’s opening scenario. As you get to know Elsa, they share more about the community and their friends with you. You learn that their active schedule does not leave much time for rest, and the same is true for their friends, who also balance jobs, sports, and schoolwork. Elsa sometimes has difficulty fitting it all in each week while getting enough sleep. They also talk about having trouble sleeping during the summer, as seasonal residents and vacationers can be noisy. Elsa feels that they have a manageable workload this week; their only major homework is a health class assignment on sexually transmitted infections. They share that the high school students have a health class meeting every other week, where they discuss health topics such as contraception, exercise, substance use, and relationship safety. Elsa appreciates the school community, which is inclusive and fosters a sense of belonging for the most part, although some students say they don’t believe people can have diverse sexual orientation or gender identities, and Elsa wonders if some classmates purposefully misgender them. But the classmates who may be doing this are few, and Elsa’s friend group spans all grades, genders, sports, interests, and other characteristics. Elsa thinks your community assessment is interesting, but they are running late to meet up with their friends at the town square for the annual Green Living Festival, which they describe as a giant yard sale of secondhand clothing and household items. The goal of the festival is for these used goods to be repurposed and stay out of the county landfill.
- With all the information you have gathered from Elsa about the community, which functional health patterns can you describe?
- What other questions would you like to ask Elsa or their friends about the community so that you could work toward developing a comprehensive community assessment?
- In your opinion, which functional health pattern is most essential for the public/community health nurse to work to address first?


5: Advanced Theories of Covalent Bonding
We have examined the basic ideas of bonding, showing that atoms share electrons to form molecules with stable Lewis structures and that we can predict the shapes of those molecules by valence shell electron pair repulsion (VSEPR) theory. These ideas provide an important starting point for understanding chemical bonding. But these models sometimes fall short in their abilities to predict the behavior of real substances. How can we reconcile the geometries of s, p, and d atomic orbitals with molecular shapes that show angles like 120° and 109.5°? Furthermore, we know that electrons and magnetic behavior are related through electromagnetic fields.
Contributors
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Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>).


5.2.1: Introduction to Bankruptcy and Overview of the 2005 Bankruptcy Act
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Learning Objectives
By the end of this section, you will be able to:
- Understand what law governs bankruptcy in the United States.
- Know the key provisions of the law.
The Purpose of Bankruptcy Law
Bankruptcy law governs the rights of creditors and insolvent debtors who cannot pay their debts. In broadest terms, bankruptcy deals with the seizure of the debtor’s assets and their distribution to the debtor’s various creditors. The term derives from the Renaissance custom of Italian traders, who did their trading from benches in town marketplaces. Creditors literally “broke the bench” of a merchant who failed to pay his debts. The term banco rotta (broken bench) thus came to apply to business failures.
In the Victorian era, many people in both England and the United States viewed someone who became bankrupt as a wicked person. In part, this attitude was prompted by the law itself, which to a greater degree in England and to a lesser degree in the United States treated the insolvent debtor as a sort of felon. Until the second half of the nineteenth century, British insolvents could be imprisoned; jail for insolvent debtors was abolished earlier in the United States. And the entire administration of bankruptcy law favored the creditor, who could with a mere filing throw the financial affairs of the alleged insolvent into complete disarray.
Today a different attitude prevails. Bankruptcy is understood as an aspect of financing, a system that permits creditors to receive an equitable distribution of the bankrupt person’s assets and promises new hope to debtors facing impossible financial burdens. Without such a law, we may reasonably suppose that the level of economic activity would be far less than it is, for few would be willing to risk being personally burdened forever by crushing debt. Bankruptcy gives the honest debtor a fresh start and resolves disputes among creditors.
History of the Bankruptcy System; Bankruptcy Courts and Judges
Constitutional Basis
The US Constitution prohibits the states from impairing the “obligation of a contract.” This means that no state can directly provide a means for discharging a debtor unless the debt has been entirely paid. But the Constitution in Article I, Section 8, does give the federal government such a power by providing that Congress may enact a uniform bankruptcy law.
Bankruptcy Statutes
Congress passed bankruptcy laws in 1800, 1841, and 1867. These lasted only a few years each. In 1898, Congress enacted the Bankruptcy Act, which together with the Chandler Act amendments in 1938, lasted until 1978. In 1978, Congress passed the Bankruptcy Reform Act, and in 2005, it adopted the current law, the Bankruptcy Abuse Prevention and Consumer Protection Act (BAPCPA). This law is the subject of our chapter.
At the beginning of the twentieth century, bankruptcies averaged fewer than 20,000 per year. Even in 1935, at the height of the Great Depression, bankruptcy filings in federal court climbed only to 69,000. At the end of World War II, in 1945, they stood at 13,000. From 1950 on, the statistics show a steep increase. During the decade before the 1978 changes, bankruptcy filings in court averaged 181,000 a year—reaching a high of 254,000 in 1975. They soared to over 450,000 filings per year in the 1980s and mostly maintained that pace until just before the 2005 law took effect (see Figure 13.1). The 2005 act—preceded by “massive lobbying largely by banks and credit card companies” CCH Bankruptcy Reform Act Briefing, “Bankruptcy Abuse Prevention and Consumer Protection Act of 2005,” April 2005, http://www.cch.com/bankruptcy/bankruptcy_04-21.pdf . —was intended by its promoters to restore personal responsibility and integrity in the bankruptcy system. The law’s critics said it was simply a way for the credit card industry to extract more money from consumers before their debts were wiped away.
Bankruptcy Action.com, www.bankruptcyaction.com/USbankstats.htm, statistics from Administrative Office of the Courts.
Bankruptcy Courts, Judges, and Costs
Each federal judicial district has a US Bankruptcy Court, whose judges are appointed by US Courts of Appeal. Unless both sides agree otherwise, bankruptcy judges are to hear only bankruptcy matters (called core proceedings). Bankruptcy trustees are government lawyers appointed by the US Attorney General. They have administrative responsibilities in overseeing the proceedings.
The filing fee for a bankruptcy is about $200, depending upon the type of bankruptcy, and the typical lawyer’s fee for uncomplicated cases is about $1,200–$1,400.
Overview of Bankruptcy Provisions
The BAPCPA provides for six different kinds of bankruptcy proceedings. Each is covered by its own chapter in the act and is usually referred to by its chapter number (see Figure 13.2).
The bankruptcy statute (as opposed to case law interpreting it) is usually referred to as the bankruptcy code . The types of bankruptcies are as follows:
- Chapter 7, Liquidation: applies to all debtors except railroads, insurance companies, most banks and credit unions, and homestead associations. 11 United States Code, Section 109(b). A liquidation is a “straight” bankruptcy proceeding. It entails selling the debtor’s nonexempt assets for cash and distributing the cash to the creditors, thereby discharging the insolvent person or business from any further liability for the debt. About 70 percent of all bankruptcy filings are Chapter 7.
- Chapter 9, Adjustment of debts of a municipality: applies to municipalities that are insolvent and want to adjust their debts. 11 United States Code, Section 109(c). (The law does not suppose that a town, city, or county will go out of existence in the wake of insolvency.)
- Chapter 11, Reorganization: applies to anybody who could file Chapter 7, plus railroads. It is the means by which a financially troubled company can continue to operate while its financial affairs are put on a sounder basis. A business might liquidate following reorganization but will probably take on new life after negotiations with creditors on how the old debt is to be paid off. A company may voluntarily decide to seek Chapter 11 protection in court, or it may be forced involuntarily into a Chapter 11 proceeding.
- Chapter 12, Adjustment of debts of a family farmer or fisherman with regular annual income. 11 United States Code, Section 109(f). Many family farmers cannot qualify for reorganization under Chapter 13 because of the low debt ceiling, and under Chapter 11, the proceeding is often complicated and expensive. As a result, Congress created Chapter 12, which applies only to farmers whose total debts do not exceed $1.5 million.
- Chapter 13, Adjustment of debts of an individual with regular income: applies only to individuals (no corporations or partnerships) with debt not exceeding about $1.3 million. 11 United States Code, Section 109(e). This chapter permits an individual with regular income to establish a repayment plan, usually either a composition (an agreement among creditors, discussed in Section 13.5, “Alternatives to Bankruptcy”) or an extension (a stretch-out of the time for paying the entire debt).
- Chapter 15, Ancillary and other cross-border cases: incorporates the United Nations’ Model Law on Cross-Border Insolvency to promote cooperation among nations involved in cross-border cases and is intended to create legal certainty for trade and investment. “Ancillary” refers to the possibility that a US debtor might have assets or obligations in a foreign country; those non-US aspects of the case are “ancillary” to the US bankruptcy case.
The BAPCPA includes three chapters that set forth the procedures to be applied to the various proceedings. Chapter 1, “General Provisions,” establishes who is eligible for relief under the act. Chapter 3, “Case Administration,” spells out the powers of the various officials involved in the bankruptcy proceedings and establishes the methods for instituting bankruptcy cases. Chapter 5, “Creditors, the Debtor, and the Estate,” deals with the debtor’s “estate”—his or her assets. It lays down ground rules for determining which property is to be included in the estate, sets out the powers of the bankruptcy trustee to “avoid” (invalidate) transactions by which the debtor sought to remove property from the estate, orders the distribution of property to creditors, and sets forth the duties and benefits that accrue to the debtor under the act.
To illustrate how these procedural chapters (especially Chapter 3 and Chapter 5) apply, we focus on the most common proceeding: liquidation (Chapter 7). Most of the principles of bankruptcy law discussed in connection with liquidation apply to the other types of proceedings as well. However, some principles vary, and we conclude the chapter by noting special features of two other important proceedings—Chapter 13 and Chapter 11.
Key Takeaway
Bankruptcy law’s purpose is to give the honest debtor a fresh start and to resolve disputes among creditors. The most recent amendments to the law were effective in 2005. Bankruptcy law provides relief to six kinds of debtors: (1) Chapter 7, straight bankruptcy—liquidation—applies to most debtors (except banks and railroads); (2) Chapter 9 applies to municipalities; (3) Chapter 11 is business reorganization; (4) Chapter 12 applies to farmers; (5) Chapter 13 is for wage earners; and (6) Chapter 15 applies to cross-border bankruptcies. The bankruptcy statutes also have several chapters that cover procedures of bankruptcy proceedings.
Exercises
- Why is bankruptcy law required in a modern capitalistic society?
- Who does the bankruptcy trustee represent?
- The three most commonly filed bankruptcies are Chapter 7, 11, and 13. Who gets relief under those chapters?


8.1: Abstract
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In traditional narratives of Minoan archaeology, the visual display of writing is usually overlooked. This chapter seeks to outline a framework for exploring the modes of display and the perception of Minoan writing by focussing on artefact categories bearing Cretan Hieroglyphic and Linear A inscriptions. Since both scripts are still undeciphered, they lend themselves to a study of their attestations as signs in the Peircean sense. Attention is therefore redirected from the written form of the specific inscriptions, the ‘signifier’ or ‘representamen’, to the physical aspects of their material supports and to the symbolic messages projected by them. Semiotic relationships that are grounded in the material properties and the performative capacities of the artefacts themselves are examined, in order to detect aspects of artefactual meaning that may not be immediately obvious from a conventional perspective. Parameters like material, size, shape, and other functional aspects of Minoan inscribed artefacts are analysed. Special emphasis is also placed on artefacts that possibly served as symbolic devices, mainly inscribed sealstones and their impressions on clay. The combination of script with images that may have constituted a visual code, and its potential for assessing literacy, is explored in the case of the Archanes Script and Cretan Hieroglyphic sealstones. Clay, metal, and stone objects carrying Linear A inscriptions of a non-administrative character are also systematically considered. The different ways scale, directionality, alignment, and the small scale of writing have informed the creation of these inscribed objects constitute one of the main questions posed. How small size could have affected the use of some inscribed objects in display events and rituals that included performance is also explored. In order to address the modes of perception of Minoan writing, the analysis relies on examining how the graphic symbols of the two scripts are arranged in the ‘graphic space’, namely the area where text is positioned and read. In this framework, directionality, alignment, and scale of the Hieroglyphic and Linear A signs are treated as indexes. Finally, the study focusses on the ways in which these parameters may have affected the experience of the inscribed artefacts by social actors, as well as the role of these objects in practices of remembrance.
How to cite this book chapter:
Flouda, G. 2013. Materiality of Minoan Writing: Modes of display and perception. In: Piquette, K. E. and Whitehouse, R. D. (eds.)
Writing as Material Practice: Substance, surface and medium.
Pp. 143-174. London: Ubiquity Press. DOI:
http://dx.doi.org/10.5334/bai.h


11.6: Summary
11.1 Avoiding the “Field of Dreams” Approach
Business models and business plans are tools involved in the strategic process of charting a path for your entrepreneurial venture. When starting a company, it’s best to avoid the Field of Dreams approach of building a business and hoping that customers will just show up. Innovation has been linked to entrepreneurship for at least nine decades. More recent theories suggest that disruptive innovations are disruptive because they disturb the underlying business model of the companies being disrupted.
11.2 Designing the Business Model
A business model, which is unique to a company, describes the rationale of how an organization creates, delivers, and captures values. A business model canvas is a popular tool used by entrepreneurs and intrapraneurs to map out and plan different components of a business model. The business model canvas addresses customer segments, customer relationships, channels (of distribution), revenue streams, value propositions, key partners, key activities, key resources, and cost structure. The lean model canvas and social business model canvas are derivations of the original business model canvas. They are designed more for tech/software/app endeavors and social entrepreneurial ventures, respectively.
11.3 Conducting a Feasibility Analysis
A feasibility study is a tool in the entrepreneurs’ toolkit that can help determine whether to proceed on a venture early on. A feasibility study is usually more in depth than a business plan and focuses on analysis of real-world numbers and projections. The most common elements of a feasibility study include a market analysis, a financial analysis, and management analysis. Feasibility studies can be used to make a “go-or-no-go” decision for a new product or business, as well as help narrow the focus of what that venture should be (what the market would potentially support).
11.4 The Business Plan
The business plan is a formal document used for long-range planning of a company’s operation that typically includes financial information, background information, and a summary of the business. Business plans can be used as internal guiding documents early in the entrepreneurial process. They also can be presentation documents for raising funds from prospective investors later in the process, complementing an investor pitch and iterative canvases. A brief business plan functions somewhat like an extended executive summary that summarizes key elements of the entire business plan, such as the business concept, financial features, and current business position. A full business plan can range from ten to twenty-five pages. It includes a business description, industry and marketing analyses, and descriptions of management, marketing, and operational and financial plans.


14.E: Thermochemistry (Exercises)
13.5: Calorimetry
Q13.5.1
After going through combustion in a bomb calorimeter a sample gives off 5,435 cal. The calorimeter experiences an increase of 4.27°C in its temperature. Using this information, determine the heat capacity of the calorimeter in kJ/°C.
Q13.5.2
Referring to the example given above about the heat of combustion, calculate the temperature change that would occur in the combustion of 1.732 g \(C_{12}H_{22}O_{12}\) in a bomb calorimeter that had the heat capacity of 3.87 kJ/°C.
Q13.5.3
Given the following data calculate the heat of combustion in kJ/mol of xylose,\(C_{5}H_{10}O_{5}\)(s), used in a bomb calorimetry experiment: mass of \(C_{5}H_{10}O_{5}\)(s) = 1.250 g, heat capacity of calorimeter = 4.728 kJ/°C, Initial Temperature of the calorimeter = 24.37°C, Final Temperature of calorimeter = 28.29°C.
Q13.5.4
Determine the heat capacity of the bomb calorimeter if 1.714 g of naphthalene, \(C_{10}H_{8}\)(s), experiences an 8.44°C increase in temperature after going through combustion. The heat of combustion of naphthalene is -5156 kJ/mol \(C_{10}H_{8}\).
Q13.5.5
What is the heat capacity of the bomb calorimeter if a 1.232 g sample of benzoic acid causes the temperature to increase by 5.14°C? The heat of combustion of benzoic acid is -26.42 kJ/g.
S13.5.1
Use equation 4 to calculate the heat of capacity:
\(q_{calorimeter} = \; heat \; capicity \; of \; calorimeter \; x \; \Delta{T}\)
5435 cal = heat capacity of calorimeter x 4.27°C
Heat capacity of calorimeter = (5435 cal/ 4.27°C) x (4.184 J/1 cal) x (1kJ/1000J) = 5.32 kJ/°C
S13.5.2
The temperature should increase since bomb calorimetry releases heat in an exothermic combustion reaction.
Change in Temp = (1.732 g \(C_{12}H_{22}O_{11}\)) x (1 mol \(C_{12}H_{22}O_{11}\)/342.3 g \(C_{12}H_{22}O_{11}\)) x (6.61 x 10³ kJ/ 1 mol \(C_{12}H_{22}O_{11}\)) x (1°C/3.87kJ) = 8.64°C
S13.5.3
[(Heat Capacity x Change in Temperature)/mass] =[ ((4.728 kJ/°C) x(28.29 °C – 24.37 °C))/1.250 g] = 14.8 kJ/g xylose
\(q_{rxn}\) = (-14.8 kJ/g xylose) x (150.13 g xylose/ 1 mol xylose) = -2.22x10³ kJ/mol xylose
S13.5.4
Heat Capacity = [(1.714 g \(C_{10}H_{8}\)) x (1 mol \(C_{10}H_{8}\)/128.2 g \(C_{10}H_{8}\)) x (5.156x10³ kJ/1 mol \(C_{10}H_{8}\))]/8.44°C = 8.17 kJ/ °C
S13.5.5
Heat Capacity = [(1.232 g benzoic acid) x (26.42 kJ/1 g benzoic acid)]/5.14°C = 6.31 kJ/ °C