1.2: Cardinality
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Often times we are interested in the number of items in a set or subset. This is called the cardinality of the set.
Cardinality
The number of elements in a set is the cardinality of that set.
The cardinality of the set \(A\) is often notated as \(|A|\) or \(n(A)\)
Example 12
Let \(A=\{1,2,3,4,5,6\}\) and \(B=\{2,4,6,8\}\)
What is the cardinality of \(B ? A \cup B, A \cap B ?\)
Solution
The cardinality of \(B\) is \(4,\) since there are 4 elements in the set.
The cardinality of \(A \cup B\) is \(7,\) since \(A \cup B=\{1,2,3,4,5,6,8\},\) which contains 7 elements.
The cardinality of \(A \cap B\) is 3 , since \(A \cap B=\{2,4,6\}\), which contains 3 elements.
Example 13
What is the cardinality of \(P=\) the set of English names for the months of the year?
Solution
The cardinality of this set is \(12,\) since there are 12 months in the year.
Sometimes we may be interested in the cardinality of the union or intersection of sets, but not know the actual elements of each set. This is common in surveying.
Example 14
A survey asks 200 people “What beverage do you drink in the morning”, and offers choices:
- Tea only
- Coffee only
- Both coffee and tea
Suppose 20 report tea only, 80 report coffee only, 40 report both. How many people drink tea in the morning? How many people drink neither tea or coffee?
Solution
This question can most easily be answered by creating a Venn diagram. We can see that we can find the people who drink tea by adding those who drink only tea to those who drink both: 60 people.
We can also see that those who drink neither are those not contained in the any of the three other groupings, so we can count those by subtracting from the cardinality of the universal set, 200.
\(200-20-80-40=60\) people who drink neither.
Example 15
A survey asks: Which online services have you used in the last month:
- Have used both
The results show 40% of those surveyed have used Twitter, 70% have used Facebook, and 20% have used both. How many people have used neither Twitter or Facebook?
Solution
Let \(T\) be the set of all people who have used Twitter, and \(F\) be the set of all people who have used Facebook. Notice that while the cardinality of \(F\) is \(70 \%\) and the cardinality of \(T\) is \(40 \%\), the cardinality of \(F \cup T\) is not simply \(70 \%+40 \%\), since that would count those who use both services twice. To find the cardinality of \(F \cup T\), we can add the cardinality of \(F\) and the cardinality of \(T\), then subtract those in intersection that we've counted twice. In symbols,
\(\mathrm{n}(F \cup T)=\mathrm{n}(F)+\mathrm{n}(T)-\mathrm{n}(F \cap T)\)
\(\mathrm{n}(F \cup T)=70 \%+40 \%-20 \%=90 \%\)
Now, to find how many people have not used either service, we're looking for the cardinality of \((F \cup T)^{c}\). since the universal set contains \(100 \%\) of people and the cardinality of \(F \cup T=90 \%\), the cardinality of \((F \cup 7)^{c}\) must be the other \(10 \%\)
The previous example illustrated two important properties
Cardinality properties
\(\mathrm{n}(A \cup B)=\mathrm{n}(A)+\mathrm{n}(B)-\mathrm{n}(A \cap B)\)
\(n\left(A^{\circ}\right)=n(U)-n(A)\)
Notice that the first property can also be written in an equivalent form by solving for the cardinality of the intersection:
\(\mathrm{n}(A \cap B)=\mathrm{n}(A)+\mathrm{n}(B)-\mathrm{n}(A \cup B)\)
Example 16
Add text here.Fifty students were surveyed, and asked if they were taking a social science (SS), humanities (HM) or a natural science (NS) course the next quarter.
\(\begin{array}{ll} \text{21 were taking a SS course} & \text{26 were taking a HM course} \\ \text{19 were taking a NS course} & \text{9 were taking SS and HM} \\ \text{7 were taking SS and NS} & \text{10 were taking HM and NS} \\ \text{3 were taking all three} & \text{7 were taking none} \end{array}\)
How many students are only taking a SS course?
Solution
It might help to look at a Venn diagram.
From the given data, we know that there are 3 students in region \(e\) and 7 students in region \(h\)
since 7 students were taking a \(S S\) and \(N S\) course, we know that \(n(d)+n(e)=7\). since we know there are 3 students in region 3 , there must be
\(7-3=4\) students in region \(d\)
Similarly, since there are 10 students taking \(\mathrm{HM}\) and \(\mathrm{NS}\), which includes regions \(e\) and \(f\), there must be
\(10-3=7\) students in region \(f\)
Since 9 students were taking \(\mathrm{SS}\) and \(\mathrm{HM}\), there must be \(9-3=6\) students in region \(b\)
Now, we know that 21 students were taking a SS course. This includes students from regions \(a, b, d,\) and \(e .\) since we know the number of students in all but region \(a,\) we can determine that \(21-6-4-3=8\) students are in region \(a\)
8 students are taking only a SS course.
Try it Now 4
One hundred fifty people were surveyed and asked if they believed in UFOs, ghosts, and Bigfoot.
\(\begin{array}{ll} \text{43 believed in UFOs} & \text{44 believed in ghosts} \\ \text{25 believed in Bigfoot} & \text{10 believed in UFOs and ghosts} \\ \text{8 believed in ghosts and Bigfoot} & \text{5 believed in UFOs and Bigfoot} \\ \text{2 believed in all three} & \text{} \end{array}\)
How many people surveyed believed in at least one of these things?
- Answer
-
Starting with the intersection of all three circles, we work our way out. since 10 people believe in UFOs and Ghosts, and 2 believe in all three, that leaves 8 that believe in only UFOs and Ghosts. We work our way out, filling in all the regions. Once we have, we can add up all those regions, getting 91 people in the union of all three sets. This leaves \(150-91=59\) who believe in none.


11.6: Steroids
- To identify the functions of steroids produced in mammals.
Triglycerides are saponifiable. That means triglycerides can react with aqueous base (alkali) to yield simpler components, such as glycerol and fatty acid salts (soap). Lipid samples extracted from cellular material, however, also contain a small but important fraction that does not react with alkali. Such a lipid is non saponifiable. The most important nonsaponifiable lipids are the steroids. These compounds include the bile salts , cholesterol, and certain hormones (such as cortisone and the sex hormones).
Steroids occur in plants, animals, yeasts, and molds but not in bacteria. They may exist in free form or combined with fatty acids or carbohydrates. All steroids have a characteristic structural component consisting of four fused rings. Chemists identify the rings by capital letters and number the carbon atoms as shown in Figure \(\PageIndex{1a}\). Slight variations in this structure or in the atoms or groups attached to it produce profound differences in biological activity.
Cholesterol
Cholesterol (Figure \(\PageIndex{1b}\)) does not occur in plants, but it is the most abundant steroid in the human body (240 g is a typical amount). Excess cholesterol is believed to be a primary factor in the development of atherosclerosis and heart disease, which are major health problems in the United States today. About half of the body’s cholesterol is interspersed in the lipid bilayer of cell membranes. Much of the rest is converted to cholic acid, which is used in the formation of bile salts. Cholesterol is also a precursor in the synthesis of sex hormones, adrenal hormones, and vitamin D.
Excess cholesterol not metabolized by the body is released from the liver and transported by the blood to the gallbladder. Normally, it stays in solution there until being secreted into the intestine (as a component of bile) to be eliminated. Sometimes, however, cholesterol in the gallbladder precipitates in the form of gallstones (Figure \(\PageIndex{2}\)). Indeed, the name cholesterol is derived from the Greek chole , meaning “bile,” and stereos , meaning “solid.”
Heart disease is the leading cause of death in the United States for both men and women. The Centers for Disease Control and Prevention reported that heart disease claimed 631,636 lives in the United States (26% of all reported deaths) in 2006.
Scientists agree that elevated cholesterol levels in the blood, as well as high blood pressure, obesity, diabetes, and cigarette smoking, are associated with an increased risk of heart disease. A long-term investigation by the National Institutes of Health showed that among men ages 30 to 49, the incidence of heart disease was five times greater for those whose cholesterol levels were above 260 mg/100 mL of serum than for those with cholesterol levels of 200 mg/100 mL or less. The cholesterol content of blood varies considerably with age, diet, and sex. Young adults average about 170 mg of cholesterol per 100 mL of blood, whereas males at age 55 may have cholesterol levels at 250 mg/100 mL or higher because the rate of cholesterol breakdown decreases with age. Females tend to have lower blood cholesterol levels than males.
To understand the link between heart disease and cholesterol levels, it is important to understand how cholesterol and other lipids are transported in the body. Lipids, such as cholesterol, are not soluble in water and therefore cannot be transported in the blood (an aqueous medium) unless they are complexed with proteins that are soluble in water, forming assemblages called lipoproteins . Lipoproteins are classified according to their density, which is dependent on the relative amounts of protein and lipid they contain.
Research on cholesterol and its role in heart disease has focused on serum levels of low-density lipoproteins (LDLs) and high-density lipoproteins (HDLs). One of the most fascinating discoveries is that high levels of HDLs reduce a person’s risk of developing heart disease, whereas high levels of LDLs increase that risk. Thus the serum LDL:HDL ratio is a better predictor of heart disease risk than the overall level of serum cholesterol. Persons who, because of hereditary or dietary factors, have high LDL:HDL ratios in their blood have a higher incidence of heart disease.
How do HDLs reduce the risk of developing heart disease? No one knows for sure, but one role of HDLs appears to be the transport of excess cholesterol to the liver, where it can be metabolized. Therefore, HDLs aid in removing cholesterol from blood and from the smooth muscle cells of the arterial wall.
LDL's transport cholesterol from the liver to the cells where they are incorporated into membrane or transformed to other steroids. Therfore high LDL levels are warning signs of atherosclerosis, the build up of cholesterol containing deposits in the arteries.
Dietary modifications and increased physical activity can help lower total cholesterol and improve the LDL:HDL ratio. The average American consumes about 600 mg of cholesterol from animal products each day and also synthesizes approximately 1 g of cholesterol each day, mostly in the liver. The amount of cholesterol synthesized is controlled by the cholesterol level in the blood; when the blood cholesterol level exceeds 150 mg/100 mL, the rate of cholesterol biosynthesis is halved. Hence, if cholesterol is present in the diet, a feedback mechanism suppresses its synthesis in the liver. However, the ratio of suppression is not a 1:1 ratio; the reduction in biosynthesis does not equal the amount of cholesterol ingested. Thus, dietary substitutions of unsaturated fat for saturated fat, as well as a reduction in consumption of trans fatty acids, is recommended to help lower serum cholesterol and the risk of heart disease.
Steroid Hormones
Hormones are chemical messengers that are released in one tissue and transported through the circulatory system to one or more other tissues. They are molecules that regulate the function of organs and tissues. One group of hormones is known as steroid hormones because these hormones are synthesized from cholesterol, which is also a steroid. There are two main groups of steroid hormones: adrenocortical hormones and sex hormones.
Adrenocorticoid Hormones
The adrenocortical hormones, such as cortisone and cortisol are produced by the adrenal gland, which is located adjacent to each kidney. They are produced from cholesterol through progesterone. These compounds are anti-inflammatory agents. Cortisol and its analogs are therefore used pharmacologically as immunosuppressants after transplant operations and in the treatment of severe skin allergies and autoimmune diseases, such as rheumatoid arthritis.
Figure \(\PageIndex{3}\) Manufacture of cortisol and cortisone starts from cholesterol.
 They are produced from cholesterol by the shortening of the hydrocarbon chain and alterations to the ring producing progesterone.  Two kinds of sex hormones are of particular importance in females: progesterone, which prepares the uterus for pregnancy and prevents the further release of eggs from the ovaries during pregnancy, and the estradiol, which are mainly responsible for the development of female secondary sexual characteristics. Both males and females produce these hormones, differing in the amounts of secreted hormones rather than in the presence or absence of one or the other. Cholesterol is the starting material for the production of these hormones.
Figure \(\PageIndex{4}\) Cholesterol is the starting material for the synthesis of the sex hormpnes.
| Hormone | Effect |
|---|---|
| regulates salt metabolism; stimulates kidneys to retain sodium and excrete potassium | |
| stimulates the conversion of proteins to carbohydrates | |
| regulates the menstrual cycle; maintains pregnancy | |
| stimulates female sex characteristics; regulates changes during the menstrual cycle | |
| stimulates and maintains male sex characteristics |
Sex hormones, both natural and synthetic, are sometimes used therapeutically. For example, a woman who has had her ovaries removed may be given female hormones to compensate. Some of the earliest chemical compounds employed in cancer chemotherapy were sex hormones. For example, estrogens are one treatment option for prostate cancer because they block the release and activity of testosterone. Testosterone enhances prostate cancer growth. Sex hormones are also administered in preparation for sex-change operations, to promote the development of the proper secondary sexual characteristics. Oral contraceptives are synthetic derivatives of the female sex hormones; they work by preventing ovulation.
Bile Salts
Bile is a yellowish green liquid (pH 7.8–8.6) produced in the liver. The most important constituents of bile are bile salts, which are sodium salts of glycocholate and taurocholate. They are synthesized from cholesterol in the liver, stored in the gallbladder, and then secreted in bile into the small intestine.
Because they contain both hydrophobic and hydrophilic groups, bile salts are amphipathic and highly effective emulsifying agents; they break down large fat globules into smaller ones and keep those smaller globules suspended in the aqueous digestive environment. Enzymes can then hydrolyze fat molecules more efficiently. Thus, the major function of bile salts is to aid in the digestion of dietary lipids.
Surgical removal is often advised for a gallbladder that becomes infected, inflamed, or perforated. This surgery does not seriously affect digestion because bile is still produced by the liver, but the liver’s bile is more dilute and its secretion into the small intestine is not as closely tied to the arrival of food.
Summary
Steroids have a four-fused-ring structure and have a variety of functions. Cholesterol is a steroid found in mammals that is needed for the rigidity of cell membranes, synthesis of bile acids, and several hormones. Bile salts are secreted into the small intestine to aid in the digestion of fats.


5.27: The Sine Distribution
The sine distribution is a simple probability distribution based on a portion of the sine curve. It is also known as Gilbert's sine distribution , named for the American geologist Grove Karl (GK) Gilbert who used the distribution in 1892 to study craters on the moon.
The Standard Sine Distribution
Distribution Functions
The standard sine distribution is a continuous distribution on \( [0, 1] \) with probability density function \(g\) given by \[g(z) = \frac{\pi}{2} \sin(\pi z), \quad z \in [0, 1] \]
- \(g\) is symmetric about \( z = \frac 1 2 \).
- \(g\) increases and then decreases with mode at \( z = \frac 1 2 \).
- \( g \) is concave downward.
Proof
From simple calculus, \( g \) is a probability density function: \( \sin(\pi x) \ge 0 \) for \( x \in [0, 1] \) and \[ \int_0^1 \sin(\pi z) dz = \frac{2}{\pi} \] The properties follow from basic calculus since \begin{align} g^\prime(z) & = \frac{\pi^2}{2} \cos(\pi z), \quad z \in [0, 1] \\ g^{\prime \prime}(z) & = -\frac{\pi^3}{2} \sin(\pi z), \quad z \in [0, 1] \end{align}
Open the Special Distribution Simulator and select the sine distribution. Run the simulation 1000 times and compare the emprical density function to the probability density function.
The distribution function \(G\) is given by \( G(z) = \frac{1}{2} [1 - \cos(\pi z)]\) for \( z \in [0, 1] \).
Proof
This follows from the PDF above and simple calculus.
The quantile function \(G^{-1}\) is given by \( G^{-1}(p) = \frac{1}{\pi} \arccos(1 - 2 p) \) for \( p \in [0, 1] \).
- The first quartile is \(q_1 = \frac{1}{3} \).
- The median is \( \frac{1}{2} \).
- The third quartile is \(q_3 = \frac{2}{3} \).
Proof
The formula for the quantile function follows immediately from the CDF above by solving \(p = G(z)\) for \(z\) in terms of \(p \in [0, 1]\).
Open the Special Distribution Calculator and select the sine distribution. Compute a few quantiles.
Moments
Suppose that \( Z \) has the standard sine distribution. The moment generating function can be given in closed form.
The moment generating function \( m \) of \( Z \) is given by \[ m(t) = \E\left(e^{t Z}\right) = \frac{\pi^2 (1 + e^t)}{2(t^2 + \pi^2)}, \quad t \in \R \]
Proof
Note first that \[ m(t) = \frac{\pi}{2} \int_0^1 e^{t z} \sin(\pi z) \, dz \] Integrating by parts with \( u = e^{t z} \) and \( dv = \sin(\pi z) dz \) gives \[ m(t) = \frac{t}{2} (1 + e^t) + \frac{t}{2} \int_0^1 e^{t z} \cos(\pi z) \, dz \] Integrating by parts again with \( u = e^{t z} \) and \( dv = \cos(\pi z) dz \) gives \[ m(t) = \frac{t}{2} (1 + e^t) - \frac{t^2}{\pi^2} m(t) \] Solving for \( m(t) \) gives the result.
The moments of all orders exist, but a general formula is complicated and involves special functions. However, the mean and variance are easy to compute.
The mean and variance of \( Z \) are
- \(\E(Z) = 1/2 \)
- \(\var(Z) = 1/4 - 2 / \pi^2\)
Proof
- We know that the mean exists since the PDF is continuous on a bounded interval. By symmetry, the mean must be \( 1/2 \).
- Integration by parts (twice) gives \[ \E(Z^2) = \int_0^1 z^2 \frac{\pi}{2} \sin(\pi z) \, dz = \frac{1}{2} - \frac{2}{\pi^2} \] The variance then follows from the usual computational formula \( \var(Z) = \E(Z^2) - [\E(Z)]^2 \).
Of course, the mean and variance could also be obtained by differentiating the MGF.
Numerically, \( \sd(Z) \approx 0.2176 \).
Open the Special Distribution Simulator and select the sine distribution. Note the position and size of the mean \(\pm \) standard deviation bar. Run the simulation 1000 times and compare the empirical mean and stadard deviation to the distribution mean and standard deviation.
The skewness and kurtosis of \(Z\) are
- \(\skw(Z) = 0\)
- \(\kur(Z) = (384 - 48 \pi^2 + \pi^4) / (\pi^2 - 8)^2\)
Proof
- The skewness is 0 by the symmetry of the distribution.
- The formula for the kurtosis follows from the usual computational formula and the first four moments: \( \E(Z) = 1/2 \), \( \E(Z^2) = 1/2 - 2 / \pi^2 \), \( \E(Z^3) = 1/2 - 3 / \pi^2 \), \( \E(Z^4) = 1/2 + 24 / \pi^4 - 6 / \pi^2 \).
Numerically, \( \kur(Z) \approx 2.1938 \).
Related Distributions
Since the distribution function and the quantile function have closed form representations, the standard sine distribution has the usual connection to the standard uniform distribution.
- If \( U \) has the standard uniform distribution then \( Z = G^{-1}(U) = \frac{1}{\pi} \arccos(1 - 2 U) \) has the standard sine distribution.
- If \( Z \) has the standard sine distribution then \( U = G(Z) = \frac{1}{2} [1 - \cos(\pi Z)] \) has the standard uniform distribution.
Part (a) of course leads to the random quantile method of simulation.
Open the random quantile simulator and select the sine distribution. Note the shape of the distribution and density functions. Run the simulation 1000 times and note the random quantiles. Compare the empirical density function to the probability density function.
Since the probability density function is continuous and is defined on a closed, bounded interval, the standard sine distribution can also be simulated using the rejection method.
Open the rejection method app and select the sine distribution. Run the simulation 1000 times and compare the empirical density function to the probability density function.
The General Sine Distribution
As with so many other
standard distributions
, the standard sine distribution is generalized by adding location and scale parameters.
Suppose that \(Z\) has the standard sine distribution. For \(a \in \R\) and \( b \in (0, \infty) \), random variable \( X = a + b Z \) has the sine distribution with location parameter \(a\) and scale parameter \(h\).
Distribution Functions
Analogies of the results above for the standard sine distribution follow easily from basic properties of the location-scale transformation. Suppose that \( X \) has the sine distribution with location parameter \( a \in \R \) and scale parameter \( b \in (0, \infty) \). So \( X \) has a continuous distribution on the interval \( [a, a + b] \).
The probability density function \( f \) of \( X \) is given by \[ f(x) = \frac{\pi}{2 b} \sin\left(\pi \frac{x - a}{b}\right), \quad x \in [a, a + b] \]
- \( f \) is symmetric about \( x = a + b / 2 \).
- \( f \) increases and then decreases, with mode \( x = a + b / 2 \).
- \( f \) is concave downward.
Proof
Recall that \[ f(x) = \frac{1}{b} g\left(\frac{x - a}{b}\right), \quad x \in \R \] where \( g \) is the standard PDF .
Pure scale transformations (\( a = 0 \) and \( b \gt 0 \)) are particularly common, since \( X \) often represents a
random angle
. The scale transformation with \( b = \pi \) gives the angle in radians. In this case the probability density function is \( f(x) = \frac{1}{2} \sin(x) \) for \( x \in [0, \pi] \). Since the radian is the standard angle unit, this distribution could also be considered the
standard one
. The scale transformation with \( b = 90 \) gives the angle in degrees. In this case, the probability density function is \( f(x) = \frac{\pi}{180} \sin\left(\frac{\pi}{90} x\right) \) for \( x \in [0, 90] \). This was Gilbert's original formulation.
In the special distribution simulator, select the sine distribution. Vary the parameters and note the shape and location of the probability density function. For selected values of the parameters, run the simulation 1000 times and compare the empirical density function to the probability density function.
The distribution function \( F \) of \( X \) is given by \[ F(x) = \frac{1}{2}\left[1 - \cos\left(\pi \frac{x - a}{b}\right)\right], \quad x \in [a, a + b] \]
Proof
Recall that \[ F(x) = G\left(\frac{x - a}{b}\right), \quad x \in \R \] where \( G \) is the standard CDF .
The quantile function \( F^{-1} \) of \( X \) is given by \[ F^{-1}(p) = a + \frac{b}{\pi} \arccos(1 - 2 p), \quad p \in (0, 1) \]
- The first quartile is \( a + b / 3\).
- The median is \( a + b / 2 \).
- The third quartile is \( a + 2 b / 3\)
Proof
Recall that \( F^{-1}(p) = a + b G^{-1}(p) \) for \( p \in (0, 1) \), where \( G^{-1} \) is the standard quantile function .
In the special distribution calculator, select the sine distribution. Vary the parameters and note the shape and location of the probability density function and the distribution function. For selected values of the parameters, find the quantiles of order 0.1 and 0.9.
Moments
Suppose again that \( X \) has the sine distribution with location parameter \( a \in \R \) and scale parameter \( b \in (0, \infty) \).
The moment generating function \( M \) of \( X \) is given by \[ M(t) = \frac{\pi^2 \left(e^{a t} + e^{(a + b) t}\right)}{2 \left(b^2 t^2 + \pi^2\right)}, \quad t \in \R \]
Proof
Recall that \( M(t) = e^{a t} m(b t) \) where \( m \) is the standard MGF .
The mean and variance of \( X \) are
- \(\E(X) = a + b / 2\)
- \(\var(X) = b^2 (1 / 4 - 2 / \pi^2)\)
Proof
By definition we can assume \( X = a + b Z \) where \( Z \) has the standard sine distribution. Using the mean and variance of \( Z \) we have
- \( \E(X) = a + b \E(Z) = a + b / 2\)
- \( \var(X) = b^2 \var(Z) = b^2 (1 / 4 - 2 / \pi^2) \)
In the special distribution simulator, select the sine distribution. Vary the parameters and note the shape and location of the mean \( \pm \) standard deviation bar. For selected values of the parameters, run the simulation 1000 times and compare the empirical mean and standard deviation to the distribution mean and standard deviation.
The skewness and kurtosis of \( X \) are
- \( \skw(X) = 0 \)
- \(\kur(X) = (384 - 48 \pi^2 + \pi^4) / (\pi^2 - 8)^2 \)
Proof
Recall that skewness and kurtosis are defined in terms of the standard score, and hence are invariant under location-scale transformations. So the skewness and kurtosis of \( X \) are the same as the skewness and kurtosis of \( Z \) .
Related Distributions
The general sine distribution is a location-scale family, so it is trivially closed under location-scale transformations.
Suppose that \( X \) has the sine distribution with location parameter \( a \in \R \) and scale parameter \( b \in (0, \infty) \), and that \( c \in \R \) and \( d \in (0, \infty) \). Then \( Y = c + d X \) has the sine distribution with location parameter \( c + a d \) and scale parameter \( b d \).
Proof
Again by definition we can take \( X = a + b Z \) where \( Z \) has the standard sine distribution. Then \( Y = c + d X = (c + a d) + (b d) Z \).


9.6: Selling and Administrative Budget
Learning Outcomes
- Create a selling and administrative expense budget
So Hupana Running Company knows all about production, and we have a good handle on how many pairs of shoes we are going to make, and how much raw materials and overhead go into each pair. But how do we get those orders? Well, there are sales commissions, marketing plans, data entry personnel, insurance, property taxes and all the other stuff that goes into operating a business.
Some of these expenses are variable, like sales commissions, and data entry personnel, while other expenses, like insurance and property taxes happen no matter how many pairs of shoes we make or sell. These are our fixed expenses. So we are going to make some assumptions so we can start work on this budget.
- Based on prior year’s actual expenses, let’s assume that the variable selling and administrative expenses are $5 per pair of shoes.
- Let’s assume that executive salaries, insurance,marketing, property taxes and depreciation are our fixed expenses.
- All of the expenses are equally distributed over the four quarters of the year.
| Hupana Running Company Selling and Administrative Expense Budget | |||||
|---|---|---|---|---|---|
| Quarter | Q1 | Q2 | Q3 | Q4 | Total |
| Budgeted unit sales | 500 | 500 | 500 | 500 | 2000 |
| Variable selling and administrative expense per case | $5 | $5 | $5 | $5 | $5 |
| Variable selling and administrative expense-total | $2,500 | $2,500 | $2,500 | $2,500 | $10,000 |
| Fixed selling and administrative expense | |||||
| Marketing | $1,500 | $1,500 | $1,500 | $1,500 | $6,000 |
| Executive salaries | $5,000 | $5,000 | $5,000 | $5,000 | $20,000 |
| Insurance | $500 | $500 | $500 | $500 | $2,000 |
| Property taxes | $250 | $250 | $250 | $250 | $1,000 |
| Depreciation expense | $500 | $500 | $500 | $500 | $2,000 |
| Total fixed selling and administrative expenses | $7,750 | $7,750 | $7,750 | $7,750 | $31,000 |
| Total selling and administrative expenses | $10,250 | $10,250 | $10,250 | $10,250 | $41,000 |
| Less depreciation | $500 | $500 | $500 | $500 | $2,000 |
| Cash disbursements for selling and administrative expense | $9,750 | $9,750 | $9,750 | $9,750 | $39,000 |
What do you notice about our selling and administrative budget? First, we use the budgeted unit sales off of the sales budget we created first! So you can start to see how everything is tied together.
Practice Questions


19.2: Body Composition Reading
Purpose
The readings will help you understand the topics covered in this week.
Directions
The rest of this chapter contains your reading for this section.
Additionally, information on these external sources will be useful to deepen your understanding:
-
Read
Health-Related Fitness Overview
Review the Module key terms and sections one and three. Focus on the body composition information.This overview document was created by Mr. Travis M. Erickson, MS, CSCS*D, Lecturer for the Appalachian State University department of Health, Leisure, and Exercise Science<EMAIL_ADDRESS>).
-
Read
ACSM_Body composition
You are not required to read the Q&A sections (pages 2 and 6). Read everything else. Notice how both this article and the previous one (#1) emphasize that Body Mass Index (BMI) is not an assessment of body composition. However, since it is a widely used screening tool it is important to know what the numbers mean in terms of classification (ex. BMI ≥ 30 = obese). -
Read Body Comp_Measurement_Encyclopedia of Lifestyle Medicine & Health (Hongu, Nobuko, Lee R. Vinson, and Scott Going. “Body Composition: Measurement Techniques.”
Encyclopedia of Lifestyle Medicine & Health
. Ed. James M. Rippe. Vol. 1. Thousand Oaks, CA: SAGE Reference, 2012. 138-143.
Gale Virtual Reference Library
.
Retrieved from
http://go.galegroup.com/ps/i.do?id=G...2894a1f451caec
)
This article will help you understand how to estimate body fat and will give you better insight into the screening tools we are using for this class: BMI and waist measurement. - Read Body Fat Distribution
This very short article introduces a new screening tool for obesity-related problems: waist-to-hip ratio. I recommend you calculate your own waist-to-hip ratio as explained in the article. This will help you understand if you are at an increased risk of health problems due to the distribution of your body fat.
- Read Female_Athlete_Triad (Ackerman, Kathryn E. “Female Athlete Triad.” Encyclopedia of Sports Medicine . Ed. Lyle J. Micheli. Vol. 2. Thousand Oaks, CA: SAGE Reference, 2011. 497-499. Gale Virtual Reference Library . Retrieved from http://go.galegroup.com/ps/i.do?id=G...9b33b3819e60f3 )
- Lifetime Fitness and Wellness. Provided by : Extended Learning Institute of Northern Virginia Community College. Located at : https://eli.nvcc.edu/ . License : CC BY: Attribution


3.1: Terms That Describe Texture
Introduction
In music, texture is how the melodic, rhythmic, and harmonic materials are combined in a composition, thus determining the overall quality of the sound in a piece. Texture is often described in regard to the density, or thickness, and range, or width, between lowest and highest pitches, in relative terms as well as more specifically distinguished according to the number of voices, or parts, and the relationship between these voices. For example, a thick texture contains many “layers” of instruments.
In musical terms, particularly in the fields of music history and music analysis, some common terms for different types of texture are:
- Monophonic
- Polyphonic
- Homophonic
- Homorhythmic
- Heterophonic
We will focus on monophony, polyphony, and homophony in this course.
Formal Terms
There are many informal terms that can describe the texture of a piece of music (thick, thin, bass-heavy, rhythmically complex, and so on), but the formal terms that are used to describe texture all describe the relationships of melodies and harmonies. Here are definitions and examples of the four main types of texture. For specific pieces of music that are good examples of each type of texture, please see below.
Monophonic
Monophonic music has only one melodic line, with no harmony or counterpoint. There may be rhythmic accompaniment, but only one line that has specific pitches. Monophonic music can also be called monophony. It is sometimes called monody, although the term “monody” can also refer to a particular type of solo song (with instrumental accompaniment) that was very popular in the 1600s.
Examples of Monophony
- One person whistling a tune
- A single bugle sounding “Taps”
- A group of people all singing a song together, without harmonies or instruments
- A fife and drum corp, with all the fifes playing the same melody
Listen for the cello performing a single melody in Bach’s Cello Suites .
Listen to the monophony in “Llibre Vermell de Montserrat [Catalonia, 14th Century]: Advocatam innocemus” by Namur Chamber Choir.
Homophonic
Homophonic music can also be called homophony. Describing homophonic music you may hear such terms as chords, accompaniment, harmony or harmonies. Homophony has one clearly melodic line; it’s the line that naturally draws your attention. All other parts provide accompaniment or fill in the chords. In most well-written homophony, the parts that are not melody may still have a lot of melodic interest. They may follow many of the rules of well-written counterpoint, and they can sound quite different from the melody and be interesting to listen to by themselves. But when they are sung or played with the melody, it is clear that they are not independent melodic parts, either because they have the same rhythm as the melody (i.e. are not independent) or because their main purpose is to fill in the chords or harmony (i.e. they are not really melodies).
Examples of Homophony
- Choral music in which the parts have mostly the same rhythms at the same time is homophonic. Most traditional Protestant hymns and most “barbershop quartet” music is in this category.
- A singer accompanied by a guitar picking or strumming chords.
- A small jazz combo with a bass, a piano, and a drum set providing the “rhythm” background for a trumpet improvising a solo.
- A single bagpipes or accordion player playing a melody with drones or chords.
In Mozart’s Piano Concerto No. 21, Andante (“Elvira Madigan”), listen for one independent melodic line accompanied by other voices or instruments.
Polyphonic
Polyphonic music can also be called polyphony, counterpoint, or contrapuntal music. If more than one independent melody is occurring at the same time, the music is polyphonic.
Examples of Polyphony
- Rounds, canons, and fugues are all polyphonic. (Even if there is only one melody, if different people are singing or playing it at different times, the parts sound independent.)
- Much Baroque music is contrapuntal, particularly the works of J.S. Bach.
- Most music for large instrumental groups such as bands or orchestras is contrapuntal at least some of the time.
- Music that is mostly homophonic can become temporarily polyphonic if an independent countermelody is added. Think of a favorite pop or gospel tune that, near the end, has the soloist “ad libbing” while the back-up singers repeat the refrain.
Bach’s Fugue in D Major is a good example of polyphony, listen for four voices occurring at the same time. Each voice is an imitation of the other, entering the piece with the same melody.
Heterophonic
A heterophonic texture is rare in Western music. In heterophony, there is only one melody, but different variations of it are being sung or played at the same time.
- Heterophony can be heard in the Bluegrass, “mountain music”, Cajun, and Zydeco traditions. Listen for the tune to be played by two instruments (say fiddle and banjo) at the same time, with each adding the embellishments, ornaments, and flourishes that are characteristic of the instrument.
- Some Middle Eastern, South Asian, central Eurasian, and Native American music traditions include heterophony. Listen for traditional music (most modern-composed music, even from these cultures, has little or no heterophony) in which singers and/or instrumentalists perform the same melody at the same time, but give it different embellishments or ornaments.
Determine the texture (monophonic, homophonic, or polyphonic) in each piece of music:
G. B. Pergolesi, Stabat Mater in F minor , arranged by J. S. Bach on the text of Psalm 51: “Tilge, Höchster, meine Sünden,” BWV 1083, c. 1746.
Contributors and Attributions
- Musical Notes texture. Authored by : duncan johnston. Located at : https://www.flickr.com/photos/duncanjohnston/4974286427/ . License : CC BY: Attribution
- Llibre Vermell de Montserrat [Catalonia, 14th Century]: Advocatam innocemus by Namur Chamber Choir . Authored by : VWIntro's channel. Located at : https://youtu.be/GbiZulU70J0 . License : All Rights Reserved . License Terms : Standard YouTube License
- Mozart, Piano Concerto No. 21, Andante (Elvira Madigan). Authored by : smalin. Located at : https://youtu.be/rKZr3ExeXUc?list=PLF03B0C0C7E1EABCB . License : All Rights Reserved . License Terms : Standard YouTube License
- Bach, Fugue in D Major, WTC II, BWV 874. Authored by : smalin. Located at : http://www.youtube.com/watch?v=m0HoK8yZS1c . License : All Rights Reserved . License Terms : Standard YouTube License
- Johann Sebastian Bachu2014Partita No. 2, Hilary Hahn. Authored by : Adagietto. Located at : https://www.youtube.com/watch?v=6KaYzgofHjc . License : All Rights Reserved . License Terms : Standard YouTube License
- Chopin Waltz Op.69 No.2 (Ashkenazy). Authored by : ardiem. Located at : https://youtu.be/cxG-kOTMgaA . License : All Rights Reserved . License Terms : Standard YouTube License


16.1: Introduction
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Think about the last concert or sporting event that you attended, observed on television, or watched on social media. What was the last piece of art that you saw in person, online, or on social media? Did you consider that your experience was likely a culmination of tens of thousands of years of human evolution? Would you consider graffiti on the sides of train cars to be art? Is a pickup game of football in the neighborhood sport? Figure 16.1 depicts a famous structure connected to sport that is now more often viewed as an art object—the Colosseum of Rome. The Colosseum served a role in ancient Roman society similar to an NFL (National Football League) stadium in contemporary American culture. Here gladiators battled animals such as lions and bears for the entertainment of a crowd of tens of thousands made up of all levels of Roman society.
As you read about the sociocultural diversity in art, music, and sports highlighted in this chapter, remember the central narrative of anthropology: Human beings have developed flexible biological and social features that have worked together in a wide variety of environmental and historical conditions to produce a diversity of cultures . Art, music, and sports have been and continue to be important elements of every culture on earth, helping to create a sense of collective identity and helping societies to hold together. Art, music, and sports both reflect the sociocultural diversity found around the world and have played roles in effecting cultural change. Art, music, and sports have shaped the evolution of societies, and the evolution of a societies has influenced art, music, and sports in turn. As you read this chapter, consider your own experiences with art, music, and sport. Consider what you are familiar with and what you appreciate. Reflect also on the art, music, and sport in your society that are not particularly interested in or moved by. Where do these preferences come from? To what degree are they individual and to what degree do they reflect connections to your culture and to the subcultures within it?


41.4: Osmoregulation and Osmotic Balance - Osmoregulators and Osmoconformers
- Compare the ability of stenohaline and euryhaline organisms to adapt to external fluctuations in salinity
Persons lost at sea without any fresh water to drink are at risk of severe dehydration because the human body cannot adapt to drinking seawater, which is hypertonic (having higher osmotic pressure) in comparison to body fluids. Stenohaline organisms, such as goldfish, can tolerate only a relatively-narrow range of salinity. About 90 percent of bony fish species can live in either freshwater or seawater, but not both. These fish are incapable of osmotic regulation in the alternate habitat.
However, a few species, known as euryhaline organisms, spend part of their lifecycle in fresh water and part in seawater. These organisms, such as the salmon, are tolerant of a relatively-wide range of salinity. They evolved osmoregulatory mechanisms to survive in a variety of aquatic environments. In relatively hypotonic (low osmotic pressure) fresh water, their skin absorbs water (see [a] in ). The fish do not drink much water and balance electrolytes by passing dilute urine while actively taking up salts through the gills. When they move to a hypertonic marine environment, the salmon lose water, excreting the excess salts through their gills and urine (see [b] in ).
Most marine invertebrates, on the other hand, may be isotonic with sea water (osmoconformers). Their body fluid concentrations conform to changes in seawater concentration. The blood composition of cartilaginous fishes, such as sharks and rays, is similar to that of bony fishes. However, the blood of sharks contains urea and trimethylamine oxide (TMAO). The shark’s blood electrolyte composition is not similar to that of seawater, but maintains isotonicity with seawater by storing urea at high concentrations. Sharks are “ureotelic” animals that secrete urea to maintain osmotic balance. TMAO stabilizes proteins in the presence of high urea levels, preventing the disruption of peptide bonds that would otherwise occur at such high levels of urea.
Contributions and Attributions
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest...ol11448/latest . License : CC BY: Attribution
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44807/latest/?collection=col11448/latest . License : CC BY: Attribution
- Osmoconformer. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Osmoconformer . License : CC BY-SA: Attribution-ShareAlike
- osmosis. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/osmosis . License : CC BY-SA: Attribution-ShareAlike
- electrolyte. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/electrolyte . License : CC BY-SA: Attribution-ShareAlike
- osmotic pressure. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/osmotic_pressure . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest...ol11448/latest . License : CC BY: Attribution
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/?collection=col11448/latest . License : CC BY: Attribution
- facilitated diffusion. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/facilitated%20diffusion . License : CC BY-SA: Attribution-ShareAlike
- passive diffusion. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/passive%20diffusion . License : CC BY-SA: Attribution-ShareAlike
- active transport. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/active%20transport . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/?collection=col11448/latest . License : CC BY: Attribution
- Transport across cell membranes. Located at : http://www.youtube.com/watch?v=RPAZvs4hvGA . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/?collection=col11448/latest . License : CC BY: Attribution
- Plasma osmolality. Provided by : Wikipedia. Located at : https://en.Wikipedia.org/wiki/Plasma_osmolality . License : CC BY-SA: Attribution-ShareAlike
- molarity. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/molarity . License : CC BY-SA: Attribution-ShareAlike
- mole. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/mole . License : CC BY-SA: Attribution-ShareAlike
- molality. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/molality . License : CC BY-SA: Attribution-ShareAlike
- osmotic pressure. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/osmotic_pressure . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/?collection=col11448/latest . License : CC BY: Attribution
- Transport across cell membranes. Located at : http://www.youtube.com/watch?v=RPAZvs4hvGA . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Concentration of solutions; part 2; moles, millimoles & milliequivalents by Professor Fink. Located at : http://www.youtube.com/watch?v=FvRe-zTUc0M . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/?collection=col11448/latest . License : CC BY: Attribution
- osmoconformer. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/osmoconformer . License : CC BY-SA: Attribution-ShareAlike
- euryhaline. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/euryhaline . License : CC BY-SA: Attribution-ShareAlike
- stenohaline. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/stenohaline . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/?collection=col11448/latest . License : CC BY: Attribution
- Transport across cell membranes. Located at : http://www.youtube.com/watch?v=RPAZvs4hvGA . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- Concentration of solutions; part 2; moles, millimoles & milliequivalents by Professor Fink. Located at : http://www.youtube.com/watch?v=FvRe-zTUc0M . License : Public Domain: No Known Copyright . License Terms : Standard YouTube license
- OpenStax College, Osmoregulation and Osmotic Balance. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44808/latest/Figure_41_01_02ab.jpg . License : CC BY: Attribution
Key Points
- Stenohaline organisms can tolerate only a relatively-narrow range of salinity.
- Euryhaline organisms are tolerant of a relatively-wide range of salinity.
- Osmoconformers are organisms that remain isotonic with seawater by conforming their body fluid concentrations to changes in seawater concentration.
Key Terms
- euryhaline : able to tolerate various saltwater concentrations
- osmoconformer : a marine organism (usually an invertebrate) that maintains its internal salinity such that it is always equal to the surrounding seawater
- stenohaline : tolerant of only a narrow range of saltwater concentrations


13.3: Conditional Probability and Intersections of Events
Events A and B are independent events if the probability of Event B occurring is the same whether or not Event A occurs.
Suppose we flipped a coin and rolled a die, and wanted to know the probability of getting a head on the coin and a 6 on the die.
Solution
We could list all possible outcomes: \(\{H1,H2,H3,H4,H5,H6,T1,T2,T3,T4,T5,T6\}\).
Notice there are \(2 \cdot 6=12\) total outcomes. Out of these, only 1 is the desired outcome \(\{H6\}\), so the probability is \(\frac{1}{12}\).
Are these events independent?
- A fair coin is tossed two times.
- The two events (1) "It will rain tomorrow in Houston" and (2) "It will rain tomorrow in Galveston” (a city near Houston).

Solution
- The probability that a head comes up on the second toss is \(\frac{1}{2}\) regardless of whether or not a head came up on the first toss, so these events are independent.
- These events are not independent because it is more likely that it will rain in Galveston on days it rains in Houston than on days it does not.
- The probability of the second card being red depends on whether the first card is red or not, so these events are not independent.
When two events are independent, the probability of both occurring is the product of the probabilities of the individual events.
If events \(A\) and \(B\) are independent, then the probability of both \(A\) and \(B\) occurring is \(P(A \text { and } B)=P(A) \cdot P(B)\)
If you look back at the coin and die example from earlier, you can see how the number of outcomes of the first event multiplied by the number of outcomes in the second event equals the total number of possible outcomes in the combined event.
In your drawer you have 10 pairs of socks, 6 of which are white, and 7 tee shirts, 3 of which are white. If you randomly reach in and pull out a pair of socks and a tee shirt, what is the probability both are white?
Solution
The probability of choosing a white pair of socks is \(\frac{6}{10}\).
The probability of choosing a white tee shirt is \(\frac{3}{7}\).
The probability of both being white is \(\frac{6}{10} \cdot \frac{3}{7}=\frac{18}{70}=\frac{9}{35}\)
A card is pulled a deck of cards and noted. The card is then replaced, the deck is shuffled, and a second card is removed and noted. What is the probability that both cards are Aces?
- Answer
-
Since the second draw is made after replacing the first card, these events are independent. The probability of an ace on each draw is \(\frac{4}{52}=\frac{1}{13},\) so the probability of an Ace on both \(\operatorname{draws}\) is \(\frac{1}{13} \cdot \frac{1}{13}=\frac{1}{169}\)
The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:
- Has a red car and got a speeding ticket
- Has a red car or got a speeding ticket.
\(\begin{array}{|l|l|l|l|}
\hline & \begin{array}{l}
\text { Speeding } \\
\text { ticket }
\end{array} & \begin{array}{l}
\text { No speeding } \\
\text { ticket }
\end{array} & \text { Total } \\
\hline \text { Red car } & 15 & 135 & 150 \\
\hline \text { Not red car } & 45 & 470 & 515 \\
\hline \text { Total } & 60 & 605 & 665 \\
\hline
\end{array}\)
Solution
We can see that 15 people of the 665 surveyed had both a red car and got a speeding ticket, so the probability is \(\frac{15}{665} \approx 0.0226\).
Notice that having a red car and getting a speeding ticket are not independent events, so the probability of both of them occurring is not simply the product of probabilities of each one occurring.
We could answer this question by simply adding up the numbers: 15 people with red cars and speeding tickets + 135 with red cars but no ticket + 45 with a ticket but no red car = 195 people. So the probability is \(\frac{195}{665} \approx 0.2932\).
We also could have found this probability by:
\(\mathrm{P}(\text {had a red car})+\mathrm{P}(\text {got a speeding ticket})-\mathrm{P}(\text {had a red car and got a speeding ticket}) =\frac{150}{665}+\frac{60}{665}-\frac{15}{665}=\frac{195}{665} \nonumber\)
Conditional Probability
Often it is required to compute the probability of an event given that another event has occurred.
What is the probability that two cards drawn at random from a deck of playing cards without replacement will both be aces?
Solution
It might seem that you could use the formula for the probability of two independent events and simply multiply \(\frac{4}{52} \cdot \frac{4}{52}=\frac{1}{169}\). This would be incorrect, however, because the two events are not independent. If the first card drawn is an ace, then the probability that the second card is also an ace would be lower because we did not replace the first ace and therefore there would only be three aces left in the deck.
Once the first card chosen is an ace, the probability that the second card chosen is also an ace is called the conditional probability of drawing an ace. In this case the "condition" is that the first card is an ace. Symbolically, we write this as:
\(P\)(ace on second draw | an ace on the first draw).
The vertical bar "|" is read as "given," so the above expression is short for "The probability that an ace is drawn on the second draw given that an ace was drawn on the first draw." What is this probability? After an ace is drawn on the first draw, there are 3 aces out of 51 total cards left. This means that the conditional probability of drawing an ace after one ace has already been drawn is \(\frac{3}{51}=\frac{1}{17}\).
Thus, the probability of both cards being aces is \(\frac{4}{52} \cdot \frac{3}{51}=\frac{12}{2652}=\frac{1}{221}\).
The probability the event \(B\) occurs, given that event \(A\) has happened, is represented as
\(P(B | A)\)
This is read as “the probability of \(B\) given \(A\)”
Find the probability that a die rolled shows a 6, given that a flipped coin shows a head.
Solution
These are two independent events, so the probability of the die rolling a 6 is \(\frac{1}{6}\), regardless of the result of the coin flip.
The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:
a) Has a speeding ticket given they have a red car
b) Has a red car given they have a speeding ticket
\(\begin{array}{|l|l|l|l|}
\hline & \begin{array}{l}
\text { Speeding } \\
\text { ticket }
\end{array} & \begin{array}{l}
\text { No speeding } \\
\text { ticket }
\end{array} & \text { Total } \\
\hline \text { Red car } & 15 & 135 & 150 \\
\hline \text { Not red car } & 45 & 470 & 515 \\
\hline \text { Total } & 60 & 605 & 665 \\
\hline
\end{array}\)
Solution
a) Since we know the person has a red car, we are only considering the 150 people in the first row of the table. Of those, 15 have a speeding ticket, so
P(ticket | red car) = \(\frac{15}{150}=\frac{1}{10}=0.1\)
b) Since we know the person has a speeding ticket, we are only considering the 60 people in the first column of the table. Of those, 15 have a red car, so
P(red car | ticket) = \(\frac{15}{60}=\frac{1}{4}=0.25\).
Notice from the last example that \(P(B | A)\) is not equal to \(P(A | B)\).
These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.
If Events \(A\) and \(B\) are not independent, then
\(P(A \text { and } B)=P(A) \cdot P(B | A)\)
If you pull 2 cards out of a deck without replacement, what is the probability that both are spades?
Solution
The probability that the first card is a spade is \(\frac{13}{52}\).
The probability that the second card is a spade, given the first was a spade, is \(\frac{12}{51}\), since there is one less spade in the deck, and one less total cards.
The probability that both cards are spades is\(\frac{13}{52} \cdot \frac{12}{51}=\frac{156}{2652} \approx 0.0588\)
If you draw two cards from a deck without replacement, what is the probability that you will get the Ace of Diamonds and a black card?
Solution
You can satisfy this condition by having Case A or Case B, as follows:
Case A) you can get the Ace of Diamonds first and then a black card or
Case B) you can get a black card first and then the Ace of Diamonds.
Let's calculate the probability of Case A. The probability that the first card is the Ace of Diamonds is \(\frac{1}{52}\). The probability that the second card is black given that the first card is the Ace of Diamonds is \(\frac{26}{51}\) because 26 of the remaining 51 cards are black. The probability is therefore \(\frac{1}{52} \cdot \frac{26}{51}=\frac{1}{102}\).
Now for Case B: the probability that the first card is black is \(\frac{26}{52}=\frac{1}{2}\). The probability that the second card is the Ace of Diamonds given that the first card is black is \(\frac{1}{51}\). The probability of Case B is therefore \(\frac{1}{2} \cdot \frac{1}{51}=\frac{1}{102}\), the same as the probability of Case 1.
Recall that the probability of \(A\) or \(B\) is \(P(\mathrm{A})+P(\mathrm{B})-P(\mathrm{A} \text { and } \mathrm{B})\). In this problem, \(P(\mathrm{A} \text { and } \mathrm{B}) = 0\) since the first card cannot be the Ace of Diamonds and be a black card. Therefore, the probability of Case A or Case B is \(\frac{1}{102}+\frac{1}{102}=\frac{2}{102}=\frac{1}{51}\). The probability that you will get the Ace of Diamonds and a black card when drawing two cards from a deck is \(\frac{1}{51}\).
In your drawer you have 10 pairs of socks, 6 of which are white. If you reach in and randomly grab two pairs of socks without replacement, what is the probability that both are white?
- Answer
-
\(\frac{6}{10} \cdot \frac{5}{9}=\frac{30}{90}=\frac{1}{3}\)
A home pregnancy test was given to women, then pregnancy was verified through blood tests. The following table shows the home pregnancy test results. Find
a) \(P\)(not pregnant | positive test result)
b) \(P\)(positive test result | not pregnant)
\(\begin{array}{|l|l|l|l|}
\hline & \begin{array}{l}
\text { Positive } \\
\text { test }
\end{array} & \text { Negative test } & \text { Total } \\
\hline \text { Pregnant } & 70 & 4 & 74 \\
\hline \text { Not Pregnant } & 5 & 14 & 19 \\
\hline \text { Total } & 75 & 18 & 93 \\
\hline
\end{array}\)
Solution
a) Since we know the test result was positive, we’re limited to the 75 women in the first column, of which 5 were not pregnant. \(P\)(not pregnant | positive test result) = \(\frac{5}{75} \approx 0.067\).
b) Since we know the woman is not pregnant, we are limited to the 19 women in the second row, of which 5 had a positive test. \(P\)(positive test result | not pregnant) = \(\frac{5}{19} \approx 0.263\).
The second result is what is usually called a false positive: A positive result when the woman is not actually pregnant.


5.1: Managing Your Research Project
- Identify reasons for outlining the scope and sequence of a research project.
- Recognize the steps of the research writing process.
- Develop a plan for managing time and resources to complete the research project on time.
- Identify organizational tools and strategies to use in managing the project.
The prewriting you have completed so far has helped you begin to plan the content of your research paper—your topic, research questions, and preliminary thesis. It is equally important to plan out the process of researching and writing the paper. Although some types of writing assignments can be completed relatively quickly, developing a good research paper is a complex process that takes time. Breaking it into manageable steps is crucial. Review the steps outlined at the beginning of this chapter.
Steps to Writing a Research Paper
- Choose a topic.
- Schedule and plan time for research and writing.
- Conduct research.
- Organize research
- Draft your paper.
- Revise and edit your paper.
You have already completed step 1. In this section, you will complete step 2. The remaining steps fall under two broad categories—the research phase of the project (steps 3 and 4) and the writing phase (steps 5 and 6). Both phases present challenges. Understanding the tasks involved and allowing enough time to complete each task will help you complete your research paper on time with a minimal amount of stress.
Planning Your Project
Each step of a research project requires time and attention. Careful planning helps ensure that you will keep your project running smoothly and produce your best work. Set up a project schedule that shows when you will complete each step. Think about how you will complete each step and what project resources you will use. Resources may include anything from library databases and word-processing software to interview subjects and writing tutors.
To develop your schedule, use a calendar and work backward from the date your final draft is due. Generally, it is wise to divide half of the available time on the research phase of the project and half on the writing phase. For example, if you have a month to work, plan for two weeks for each phase. If you have a full semester, plan to begin research early and to start writing by the middle of the term. You might think that no one really works that far ahead, but try it. You will probably be pleased with the quality of your work and with the reduction in your stress level.
As you plan, break down major steps into smaller tasks if necessary. For example, step 3, conducting research, involves locating potential sources, evaluating their usefulness and reliability, reading, and taking notes. Defining these smaller tasks makes the project more manageable by giving you concrete goals to achieve.
Jorge had six weeks to complete his research project. Working backward from a due date of May 2, he mapped out a schedule for completing his research by early April so that he would have ample time to write. Jorge chose to write his schedule in his weekly planner to help keep himself on track.
Review Jorge’s schedule. Key target dates are shaded. Note that Jorge planned times to use available resources by visiting the library and writing center and by meeting with his instructor.
Figure \(\PageIndex{1}\)
- Working backward from the date your final draft is due, create a project schedule. You may choose to write a sequential list of tasks or record tasks on a calendar.
- Check your schedule to be sure that you have broken each step into smaller tasks and assigned a target completion date to each key task.
- Review your target dates to make sure they are realistic. Always allow a little more time than you think you will actually need.
Plan your schedule realistically, and consider other commitments that may sometimes take precedence. A business trip or family visit may mean that you are unable to work on the research project for a few days. Make the most of the time you have available. Plan for unexpected interruptions, but keep in mind that a short time away from the project may help you come back to it with renewed enthusiasm. Another strategy many writers find helpful is to finish each day’s work at a point when the next task is an easy one. That makes it easier to start again.
Writing at Work
When you create a project schedule at work, you set target dates for completing certain tasks and identify the resources you plan to use on the project. It is important to build in some flexibility. Materials may not be received on time because of a shipping delay. An employee on your team may be called away to work on a higher-priority project. Essential equipment may malfunction. You should always plan for the unexpected.
Staying Organized
Although setting up a schedule is easy, sticking to one is challenging. Even if you are the rare person who never procrastinates, unforeseen events may interfere with your ability to complete tasks on time. A self-imposed deadline may slip your mind despite your best intentions. Organizational tools—calendars, checklists, note cards, software, and so forth—can help you stay on track.
Throughout your project, organize both your time and your resources systematically. Review your schedule frequently and check your progress. It helps to post your schedule in a place where you will see it every day. Both personal and workplace e-mail systems usually include a calendar feature where you can record tasks, arrange to receive daily reminders, and check off completed tasks. Electronic devices such as smartphones have similar features.
Organize project documents in a binder or electronic folder, and label project documents and folders clearly. Use note cards or an electronic document to record bibliographical information for each source you plan to use in your paper. Tracking this information throughout the research process can save you hours of time when you create your references page.
Revisit the schedule you created in Exercise 1. Transfer it into a format that will help you stay on track from day to day. You may wish to input it into your smartphone, write it in a weekly planner, post it by your desk, or have your e-mail account send you daily reminders. Consider setting up a buddy system with a classmate that will help you both stay on track.
Some people enjoy using the most up-to-date technology to help them stay organized. Other people prefer simple methods, such as crossing off items on a checklist. The key to staying organized is finding a system you like enough to use daily. The particulars of the method are not important as long as you are consistent.
Anticipating Challenges
Do any of these scenarios sound familiar? You have identified a book that would be a great resource for your project, but it is currently checked out of the library. You planned to interview a subject matter expert on your topic, but she calls to reschedule your meeting. You have begun writing your draft, but now you realize that you will need to modify your thesis and conduct additional research. Or you have finally completed your draft when your computer crashes, and days of hard work disappear in an instant.
These troubling situations are all too common. No matter how carefully you plan your schedule, you may encounter a glitch or setback. Managing your project effectively means anticipating potential problems, taking steps to minimize them where possible, and allowing time in your schedule to handle any setbacks.
Many times a situation becomes a problem due only to lack of planning. For example, if a book is checked out of your local library, it might be available through interlibrary loan, which usually takes a few days for the library staff to process. Alternatively, you might locate another, equally useful source. If you have allowed enough time for research, a brief delay will not become a major setback.
You can manage other potential problems by staying organized and maintaining a take-charge attitude. Take a minute each day to save a backup copy of your work on a portable hard drive. Maintain detailed note cards and source cards as you conduct research—doing so will make citing sources in your draft infinitely easier. If you run into difficulties with your research or your writing, ask your instructor for help, or make an appointment with a writing tutor.
Identify five potential problems you might encounter in the process of researching and writing your paper. Write them on a separate sheet of paper. For each problem, write at least one strategy for solving the problem or minimizing its effect on your project.
Writing at Work
In the workplace, documents prepared at the beginning of a project often include a detailed plan for risk management. When you manage a project, it makes sense to anticipate and prepare for potential setbacks. For example, to roll out a new product line, a software development company must strive to complete tasks on a schedule in order to meet the new product release date. The project manager may need to adjust the project plan if one or more tasks fall behind schedule.
- To complete a research project successfully, a writer must carefully manage each phase of the process and break major steps into smaller tasks.
- Writers can plan a research project by setting up a schedule based on the deadline and by identifying useful project resources.
- Writers stay focused by using organizational tools that suit their needs.
- Anticipating and planning for potential setbacks can help writers avoid those setbacks or minimize their effect on the project schedule.


4.5: Interpreting the Law
- Discuss changing interpretations of legal protections of diversity
As illustrated above, laws are subject to interpretation—both expansion and contraction. For example, an EEOC notice [1] emphasizes that their interpretation of the Title VII reference to “sex” is broadly applicable to gender, gender identity and sexual orientation. And further, that “these protections apply regardless of any contrary state or local laws.” This interpretation is consistent with Executive Orders issued by the Obama administration that extended discrimination protection based on sexual orientation or gender identity to federal workers and federal contractor employees. However, the Trump administration is seeking to reverse that interpretation and the associated protections.

In a brief submitted to the court on August 23, 2019, the Department of Justice argues that federal employment law doesn’t protect workers from discrimination based on sexual orientation. A SHRM article notes that “the department’s lawyers said that the ordinary meaning of “sex” is biologically male or female and doesn’t include sexual orientation.” [2] Major companies, like the EEOC, are fighting to retain protections. In a SHRM article, attorney Allen Smith states that “more than 200 businesses signed a brief on July 2 calling on the Supreme Court to rule that Title VII of the Civil Rights Act of 1964 prohibits discrimination based on sexual orientation and gender identity.” [3] The Supreme Court has agreed to hear two cases, one from a federal appeals court in New York that found that discrimination against gay men and lesbians is a form of sex discrimination and one from a court in Georgia that came to the opposite conclusion. The justices also agreed to decide the separate question of whether Title VII bars discrimination against transgender people.
The New York Times reports that in a minor case, “Justice Neil M. Gorsuch wrote that courts should ordinarily interpret statutes as they were understood at the time of their enactment.” Justice Ruth Bader Ginsburg held that “Congress may design legislation to govern changing times and circumstances.” Adding, from a previous opinion, “Words in statutes can enlarge or contract their scope as other changes, in law or in the world, require their application to new instances or make old applications anachronistic.” [4]
Law is interpreted (and, by extension, enforced) differently by different Administrations and courts. As Zadie Smith observed: “Progress is never permanent, will always be threatened, must be redoubled, restated and reimagined if it is to survive.”
- "What Your Should Know About EEOC and the Enforcement Protections of LGBT Workers." U.S. Equal Employment Opportunity Commission. Accessed September 14, 2019. ↵
- Nagele-Piazza, Lisa. "DOJ Asks Supreme Court to Find Workplace Bias Law Doesn't Cover Sexual Orientation." Society for Human Resource Management. August 26, 2019. Accessed September 14, 2019. ↵
- Smith, Allen. "Companies Urge High Court to Ban LGBTQ Discrimination." Society for Human Resources Management. July 2, 2019. Accessed September 14, 2019. ↵
- Liptak, Adam. "Supreme Court to Decide Whether Landmark Civil Rights Law Applies to Gay and Transgender Workers." The New York Times.April 22, 2019. Accessed September 14, 2019. ↵


8.1: Introduction
Introduction
With the advent of the twenty-first century, communication, transportation, and supply chain integration created a truly global world. No one country is isolated from events happening throughout the world. One of the major issues confronting the global population is climate change and its effects on the environment and how people and animals survive. Global warming has changed the climate on the planet as desert areas are hotter and uninhabitable for both humans and animals, arctic regions are melting, and extreme weather events are common. Land and air are polluted by the toxic waste of modern mining, manufacturing, and other industries. Some industries contribute to deforestation as forests in the Amazon or Indonesia are cleared for commercial agriculture. No other civilization had interfered with nature and the environment as current civilizations. These issues have also changed how many artists work who now use modern waste materials for their art, recycling the waste they found.
The new millennium brought multiple inventions into our lives, changing how we live. Nanotechnology and miniaturization allowed the computer industry to create smaller devices and technology into everything, including our refrigerators. The iPhone brought worldwide communications to everyone. Some countries did not have any phone systems, especially in rural areas. The portable, satellite-based phone gave everyone the potential to talk and search the world's information. Space exploration expanded with the International Space Station, missions landing on Mars, or personal rockets carrying ordinary citizens into space. Thousands of satellites roam through space, delivering the internet and social media, tracking weather, or mapping the changes to the earth. The discovery of human genomes led to transformational innovation in genetics and medicine. These inventions brought changes worldwide.
At the beginning of the year 2000, significant social changes occurred. Women gained more rights in different countries, and over the last twenty years, the number of female leaders in the world has doubled. Many countries have given women far more rights and equality, including the ability to earn equal wages. Unfortunately, women still have few rights in some places, live in poverty, and suffer physical and mental abuse. The fight for equality also helped the rise of female artists throughout the world and their breakthrough into the art markets. Same-sex marriage also became a legal concept in a growing number of countries. Terrorism continues to be a problem throughout the world, either individual terrorists' actions or country-on-country incursions.
Contemporary Architectural Materials
Contemporary architecture embraced natural light, open interiors, natural features, and eco-friendly materials. Designs might be asymmetric, have cantilevered sections, or large planes of glass and aluminum. These requirements lead to creative inventiveness for materials to support the new designs. Cement is an essential part of the materials for buildings, and the unusual shapes and need for support brought new types of fast-drying and self-healing cement. If water enters a crack in the cement, bacteria is reactivated, mixes with the cement, and excretes material to heal the crack. Bamboo is a renewable resource, a strong and resilient material used in many buildings. Aluminum is made with new technology and appears as glass yet maintains the strength of steel. Some companies are making bricks that absorb air pollution and are strong. Other materials are made from recycled wood or steel materials, reducing the impact on the environment.
Contemporary African Art Medium
Contemporary African Art is, by its very nature, fluid, self-motivated, and resists classification. [1] Art in Africa is not based on a singular concept, as art is created from local circumstantial parameters throughout the continent. Some artists have never left their country, while others have traveled and been educated in different methods. Each artist's journey into and through the art world is different, and the medium they use is as varied as the artists themselves. As different countries became independent from their colonial masters, artists were also released from traditional European painting styles. They also had to adjust to the effects of globalization on their environments. In many African countries, recycled materials became a significant art medium, allowing artists to create unique concepts based on the pollution of the twenty-first century. They produced a unique art form.
Contemporary Figurative Art Concepts
Photography captured the figure in its perfect form, rendering the exactness of the eye of the camera. The new contemporary figures now used the body in exaggeration, creating some parts overextended, extra-large, or disjointed yet portraying the body in a realistic setting. They used the body set in the form of what it might feel like in a situation instead of what it looked like. The figure may not exist in the painted image; however, the artist is depicting a narrative of what might happen the emotions the figure brings to the artwork. The works might be contradictory, depicting different expressions for experimentation. The contemporary figurative artists create a sense of existence itself, the feelings of identity, and the political events around the world.
[1] Retrieved from https://www.contemporary-african-art...rican-art.html


2.9: Depression
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Depression, or depressive disorders, is a leading cause of disability in the United States as well as worldwide. It affects an estimated 9.5 percent of American adults in a given year. Nearly twice as many women as men have depression. Depression is more than just being in a bad mood or feeling sad. Everyone experiences these feelings on occasion, but that does not constitute depression. Depression is actually not a single disease; there are three main types of depressive disorders. They are major depressive disorder, dysthymia, and bipolar disorder (manic-depression).
While some of the symptoms of depression are common during a passing “blue mood,” major depressive disorder is diagnosed when a person has five or more of the symptoms nearly every day during a two-week period. Symptoms of depression include a sad mood, a loss of interest in activities that one used to enjoy, a change in appetite or weight, oversleeping or difficulty sleeping, physical slowing or agitation, energy loss, feelings of worthlessness or inappropriate guilt, difficulty concentrating, and recurrent thoughts of death or suicide.
When people have depression, their lives are affected severely: they have trouble performing at work or school, and they aren’t interested in normal family and social activities. In adults, an untreated major depressive episode lasts an average of nine months. At least half of the people who experience an episode of major depression will have another episode of depression at some point.


5.3: Local Level
The idea of jails has a long history, and the historical roots of American jails are in the "gaols" of feudal England. Sheriffs operated these early jails, and their primary purpose was to hold accused persons awaiting trial. This English model was brought over to the Colonies, but the function remained the same. In the 1800s, jails began to change in response to the penitentiary movement. Their function was extended to housing those convicted of minor offenses and sentenced to short terms of incarceration. They were also used for other purposes, such as holding the mentally ill and vagrants. The advent of a separate juvenile justice system and the development of state hospitals alleviated the burden of taking care of these later categories.
Today's jails are critical components of local criminal justice systems. They are used to address the need for secure detention at various points in the criminal justice process. Jails typically serve several law enforcement agencies in the community, including local law enforcement, state police, wildlife conservation officers, and federal authorities. Jails respond to many needs in the criminal justice system and play an integral role within every tier of American criminal justice. These needs are ever changing and influenced by the policies, practices, and philosophies of the many different users of the jail. Running a jail is a tough business, usually undertaken by a county sheriff. Often, much of the Sheriff's authority is delegated to a jail administrator.
Running a jail is such a complicated endeavor partly because jails serve an extremely diverse population. Unlike prisons where inmate populations are somewhat homogenous, fails hold vastly different individuals. Jails hold both men and women, and both children and adults. Most state prisoners are serious offenders, whereas jails old both serious offenders as well as minor offenders who may be vulnerable to predatory criminals. Those suffering from mental illness, alcoholism, and drug addiction often find themselves in jail. It is in this environment that jail staff must accomplish the two major functions of jails: Intake and Custody.
Booking and Intake
The booking and intake function of jails serves a vital public safety function by providing a secure environment in which potentially dangerous persons can be assessed, and the risk these individuals pose the public can be determined.
Custody
The second major function of jails is the idea of custody. That is, people are deprived of their liberty for various reasons. The two most common of these reasons are pretrial detention and punishment .
Pretrial Detention
A major use of modern jails is what is often referred to as pretrial detention . In other words, jails receive accused persons pending arraignment and hold them awaiting trial, conviction, or sentencing. More than half of jail inmates are accused of crimes and are awaiting trial. The average time between arrest and sentencing is around six months. Jails also readmit probation and parole violators and absconders, holding them for judicial hearings. The major purpose of pretrial detention is not to punish offenders, but to protect the public and ensure the appearance of accused persons at trial.
According to the Bureau of Justice Statistics, there are around 3,300 jails currently in operation within the United States. This large number points to a very important fact: Jails are primarily a local concern. Jails (and detention centers ) are facilities designed to safely and securely hold a variety of criminal offenders, usually for a short period. The wide variety of offenders comes from the fact that jails have dual roles. They hold criminal defendants waiting on processing by the criminal justice system, and they hold those convicted of crimes and sentenced to a jail term. In addition, jails hold prisoners for other agencies, such as state departments of correction, until bed space becomes available in a state prison.
Figure 5.2 Jail Cell. Free to use by Pixaby
The size of jails can vary widely depending on the jurisdiction the facility serves. Both geographic and legal jurisdiction must be considered. The single most important determinant of jail size is population density. The more people a given jurisdiction has, the more jail inmates they are likely to have. Many rural jails are quite small, but America's largest population centers tend to have massive jail complexes. Most counties and many municipalities operate jails, and a few are operated by federal and other non-local agencies. There has been a trend for small, rural jurisdictions to combine their jails into regional detention facilities. These consolidated operations can increase efficiency, security, and better ensure prisoners' rights.
Punishment
A primary function of jails is to house criminal defendants after arrest. Within a very narrow window of time, the arrestee must appear before a judge. The judge will consider the charges against the defendant and the defendant's risk of flight when determining bail. The judge may decide to remand the defendant to the custody of the jail until trial, but this is rare. Most often, pretrial release will be granted. The arrestees may be required to pay a certain amount of money to ensure their appearance in court, or they may be released on their own recognizance.
As a criminal sanctioning option, jails provide a method of holding offenders accountable for criminal acts. Jails house offenders that have been sentenced to a jail term for misdemeanor offenses, usually for less than one year. There are many ways that jail sentences can be served, depending largely on the laws and policies of the particular jurisdiction. A central goal of incarceration as punishment in the criminal justice system is the philosophical goal of deterrence . Many believe that jail sentences discourage offenders from committing future criminal acts (specific deterrence) and to potential criminals about the possible costs of crime (general deterrence). Rehabilitation and reintegration are sometimes considered secondary goals of incarceration. These goals are not usually deemed amenable to the jail environment, and few programs designed to meet these goals exist. Many local jails do make a modest effort to provide inmates with opportunities for counseling and change to deter future criminal behavior, but always within the constraints of scant resources.
Miscellaneous Functions
Jails in some jurisdictions are responsible for transferring and transporting inmates to federal, state, or other authorities.  Jails also hold people for a variety of government purposes; they hold individuals wanted by the armed forces, for protective custody of individuals who may not be safe in the community, for those found in contempt of court, and witnesses for the courts. Jails often hold state and federal inmates due to overcrowding in prison facilities. Jails are commonly tasked with community-based sanctions, such as work details engaged in public services.
Jail Populations
Arrestees often arrive at the jail with myriad many problems. Substance abuse, alcohol abuse, and mental illness often mean that jail inmates are not amenable to complying with the directions of jail staff. Many have medical problems, psychological problems, and emotional problems. Inmates can display the full gambit of human emotions: fail staff may see fear, anxiety, anger, and depression every day. Behaviors often mirror emotional state, and at times staff must deal with noncompliant, suicidal, or violent inmates. While inmates are in custody, the jail is responsible for their health and wellbeing.
Jails function in a role as a service provider for the rest of the criminal justice community. Jail administrators have very little discretion in who goes to jail and how long they remain in custody. Law and policy play a big role in dictating who goes to jail, as do the discretionary decisions of probation and parole officers, law enforcement, and judges. Prevalent community attitudes are also important, because voters can place pressure on law enforcement and the courts to make more arrests and prosecute more offenders. When this happens, more people end up in jail.