7.7B: Ischium
The ischium forms the lower and posterior portion of the hip bones of the pelvis.
- Describe the ischium
Key Points
- The ischium articulates with the ilium and the pubis.
- The ischium is divided into the body, superior ramus of the ischium, and inferior ramus of the ischium.
- The ischial tuberosity, which supports weight when sitting, is located on the ischium.
Key Terms
- ischium : The most inferior of the three bones that make up each side of the pelvis.
The ischium forms the lower and back part of the hip bone. Situated below the ilium and behind the pubis, the superior portion of this bone forms approximately one third of the acetabulum, which articulates with the femoral head to form the hip joint.
Lateral view of ischium : Left hip joint, opened by removing the floor of the acetabulum from within the pelvis. The ischium is labeled at the bottom left of the ilium.
The ischium is divisible into three portions; the body, and the superior and inferior rami.
The body contains a prominent spine that is the origin for the gemellus superior muscle. Two indentations run parallel to the spine—superiorly,the greater sciatic notch and, inferiorly, the lesser sciatic notch, through which key nervous and vascular vessels pass.
The superior ramus of the ischium extends inferiorly and posteriorly from the body. It is the partial origin for the obturator internus and obturator externus muscles.
Posteriorly the ramus forms a large swelling termed the tuberosity of the ischium, or ischial tuberosity, which supports weight while sitting and is the origin for the gemellus inferior and adductor magnus muscles.
Dorsally, the ramus contributes to the obturator foramen, a large opening in the pelvis through which key nervous and vascular vessels pass.
The inferior ramus of the ischium is thin and flattened and ascends from the superior ramus of the ischium to join the inferior ramus of the pubis. It is the partial origin for the gracillis and adductor magnus muscles.
Ischium : The ischium is located below the ilium and behind the pubis.


Connect the purpose of school-age care to important aspects of planning curriculum for school-aged children
Explain how the needs of school-aged children and desired outcomes are met through developmentally curriculum
Justify the importance of collaboration between educators and children, families, and communities
Examine the role of reflective practice in curriculum planning
Define what a holistic approach to curriculum is
Analyze the value of play in school-age care and curriculum
Identify ways to plan for the different types and aspects of play
Discuss the importance of intentionality
List considerations for planning the environment for school-aged children
Summarize the role of diversity and the importance of cultural competence in school-aged programs
Describe how quality curriculum planning meets the five outcomes
Relate each step of the curriculum planning process
Differentiate the different roles of the educator
During the school-age years (kindergarten to grade 8), children may be cared for during out-of-school times in a variety of school-age programming. The focus, philosophy, staffing, and location of those programs will vary greatly. This chapter will briefly discuss some information that can be used to create high quality curriculum for children in school-age programming.
But before we get to that, let’s look at what school-age children are like. Meet these school-aged children:


4.6: How good is the proportion?
Proportions are frequent in the data analysis, especially for categorical variables. How to check how well the sample proportion corresponds with population proportion?
Here is an example. In hospital, there was a group of 476 patients undergoing specific treatment and 356 among them are smokers (this is the old data). In average, proportion of smokers is slightly less than in our group (70% versus 75%, respectively). To check if this difference is real, we can run the proportions test :
Code \(\PageIndex{1}\) (R):
(We used two.sided option to check both variants of inequality: larger and smaller. To check one of them (“one tail”), we need greater or less \(^{[1]}\).)
Confidence interval is narrow. Since the null hypothesis was that “true probability of is equal to 0.7” and p-value was less than 0.05, we reject it in favor to alternative hypothesis, “true probability of is not equal to 0.7”. Consequently, proportion of smokers in our group is different from their proportion in the whole hospital.
Now to the example from foreword. Which candidate won, A or B? Here the proportion test will help again\(^{[2]}\):
Code \(\PageIndex{2}\) (R):
According to the confidence interval, the real proportion of people voted for candidate A varies from 100% to 47%. This might change completely the result of elections!
Large p-value suggests also that we cannot reject the null hypothesis. We must conclude that “true p is not greater then 0.5”. Therefore, using only that data it is impossible to tell if candidate A won the elections.
This exercise is related with phyllotaxis (Figure 4.7.1), botanical phenomenon when leaves on the branch are distributed in accordance with the particular rule. Most amazingly, this rule ( formulas of phyllotaxis ) is quite often the Fibonacci rule , kind of fraction where numerators and denominators are members of the famous Fibonacci sequence. We made R function Phyllotaxis() which produces these fractions:
Code \(\PageIndex{3}\) (R):
In the open repository, there is a data file phyllotaxis.txt which contains measurements of phyllotaxis in nature. Variables N.CIRCLES and N.LEAVES are numerator and denominator, respectively. Variable FAMILY is the name of plant family. Many formulas in this data file belong to “classic” Fibonacci group (see above), but some do not. Please count proportions of non-classic formulas per family, determine which family is the most deviated and check if the proportion of non-classic formulas in this family is statistically different from the average proportion (calculated from the whole data).


19.2: The Particle in a Box
Now we can consider a particle constrained to move in a single dimension, under the influence of a potential \(V(x)\) which is zero for \(0 \leq x \leq a\) and infinite elsewhere. Since the wavefunction is not allowed to become infinite, it must have a value of zero where \(V(x)\) is infinite, so \(\psi(x)\) is nonzero only within \([0,a]\). The Schrödinger equation is thus:
\[ - \dfrac{\hbar^2}{2m} \dfrac{d^2\psi}{dx^2} = E \psi(x) \qquad 0 \leq x \leq a. \label{20.2.1} \]
In other words, inside the box \(\psi(x)\) describes a free particle, but outside the box \(\psi(x)=0\). Since the Schrödinger equation involves derivatives, the function that solves it, \(\psi(x)\), must be everywhere continuous and everywhere continuously differentiable. This fact means that the value of the wave function at the two extremes must be equal to zero:
\[ \psi(0)=\psi(a)=0. \label{20.2.2} \]
Inside the box we can use Euler’s formula to write the wave function as a linear combination of the positive and negative solutions:
\[ \psi(x)=A \exp(\pm ix)=A \sin kx + B \cos kx, \label{20.2.3} \]
where \(A\) and \(B\) are constants that we need to determine using the two constraints in Equation \ref{20.2.2}. For \(B\) it is straightforward to see that:
\[ \psi(0)= 0 + B =0 \; \implies \; B=0. \label{20.2.4} \]
For \(A\) we have:
\[ \psi(a)= A\sin ka = 0, \label{20.2.5} \]
which is trivially solved by \(A=0\), or by the more interesting condition of \(ka=n\pi\). The trivial solution corresponds to a wave function uniformly equal to zero everywhere. This wave function is uninteresting, since it describes no particles in no boxes. The second set of solutions, however, is very interesting, since we can write it as:
\[ \psi_n(x)= A\sin\left(\dfrac{n\pi x}{a} \right)\quad n=1,2,\ldots,\infty, \label{20.2.6} \]
which represents an infinite set of functions, \(\psi_n(x)\), determined by a positive integer number \(n\), called quantum number . Since these functions solve the TISEq, they are also called eigenfunctions , but they are not a continuous set, unlike in the previous case. To calculate the energy eigenvalues, we can replace \ref{20.2.6} into Equation \ref{20.2.1}, to obtain:
\[ E_n = \dfrac{h^2 n^2}{8 m a^2} \quad n=1,2,\ldots,\infty. \label{20.2.7} \]
A few interesting considerations can be made from the results of Equation \ref{20.2.7}. First, although there is an infinite number of acceptable values of the energy (eigenvalues), these values are not continuous. Second, the lowest value of the energy is not zero, and it depends on the size of the box, \(a\), since:
\[ E_1 = \dfrac{h^2 }{8 m a^2} \neq 0. \label{20.2.8} \]
This value is called zero-point energy (ZPE) , and is a purely quantum mechanical effect. Notice that we did not solve for the constant \(A\). This task is not straightforward, and it can be achieved by requiring the wave function to describe one particle exclusively (we will come back to this task after chapter 23 ). Extending the problem to three dimensions is relatively straightforward, resulting in a set of three separate quantum numbers (one for each of the 3-dimensional coordinate \(n_x,n_y,n_z\)).


4.6: References
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18.6: Basic Constructs in Rotter's Social Learning Theory
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Rotter’s early research focused on the need to understand human behavior and personality so that clinical psychologists might effectively help their patients. In the preface to Social Learning & Clinical Psychology , Rotter wrote:
…the practice of clinical psychology in many instances is unsystematic and confused when viewed from logical or rigorous scientific viewpoints. This confusion, however, is not a necessary condition but the result of the failure of the clinical psychologists’ training program to translate and relate the basic knowledge of experimental and theoretical psychology into the practical situations of the clinic, the hospital, and the school… (pg. viii; Rotter, 1954)
Given his emphasis on clinical psychology, Rotter focused on the clinician’s ability to predict behavior. According to Rotter, social learning theory assumes that the unit of investigation for the study of personality is the interaction between the individual and their meaningful environment. Although personality has unity, the individual’s experiences influence each other. As a result, personality is continuously changing, since each person is always having new experiences. However, personality is also stable in some respects, since previous experiences influence new learning. Given the complexity of each individual, Rotter believed that in order to make reasonable predictions about behavior it was necessary to examine four kinds of variables: behavior potential, expectancy, reinforcement value, and the psychological situation (Rotter, 1954, 1964, 1972; Rotter & Hochreich, 1975).
Behavior potential refers to the likelihood of a certain behavior occurring in the context of specific potential reinforcement. For example, in order to earn good grades a student can rely on any number of possible behaviors, such as studying, cheating, skipping class to avoid a bad grade, etc. Each potential behavior can only be described as more or less likely than other potential behaviors, and included as potential behaviors are psychological reactions such as thoughts, emotions, and even defense mechanisms. Expectancy is defined as the probability held by the individual that reinforcement will follow one’s chosen behavior. Although Rotter preferred to avoid the concept that expectancy is subjective, he acknowledged that an element of subjectivity is involved. Regardless, it is the individual’s point of view, their expectations in a given situation, that are more important for predicting behavior than the realistic probability of a chosen behavior resulting in an expected reinforcement. Reinforcement value, quite simply, refers to the preference for a given reinforcer. To use Rotter’s own example, most people would consistently choose to be paid $10 dollars an hour rather than $1 an hour, if it were simply their choice. Finally, there is the psychological situation. According to Rotter, it is not enough to say that to each individual a given situation might seem different. In order to address the situation in more objective terms, psychologists need to identify a variety of cues within the situation. In an objective sense, consequently, different people can be described as attending to different specific cues in the environment (Rotter, 1954, 1972; Rotter & Hochreich, 1975).
Although Rotter broke new ground in this approach to the study of social learning theory, he did not entirely abandon the use of mathematical formulae similar to those of Dollard and Miller. Rotter proposed the following basic formula for predicting goal-directed behavior:
BP x ,S1Ra = /(E x,RaS1 & RV a ,S1 )
Although this formula appears complicated at first glance, it is relatively straightforward. The potential for behavior x (BPx) to occur in situation 1 with potential reinforcement a (S1Ra) is a function (/) of the expectancy (E) that reinforcement a will follow behavior x in situation 1 (x,RaS1) and the reinforcement value (RV) of reinforcement a in situation 1 (a,S1) (Rotter, 1954; Rotter & Hochreich, 1975). In other words, we are most likely to choose the behavioral option that we realistically expect will result in the most favorable outcome in our current situation.
discussion question \(\PageIndex{1}\)
Rotter believed that both the expectancy of reward and the perceived value of that reward were essential in determining whether an individual engaged in a particular behavior. Have you ever found yourself doing something even though you did not expect to get anything for your efforts?  In such situations, how long did you continue your behavior, and how did you feel about it?
Locus of Control
One of the most important generalized expectancies underlying behavior, and perhaps Rotter’s best known concept, is referred to as internal versus external control of reinforcement (commonly known as locus of control ):
People are known to differ in their belief that what happens to them is the result of their own behaviors and attributes (internal control) versus the result of luck, fate, chance, or powerful others (external control). Clearly, persons who believe or expect that they can control their own destinies will behave differently, in many situations, than those who expect that their outcomes are controlled by other people or determined by luck. (pg. 105; Rotter & Hochreich, 1975)
Rotter pointed out that almost all psychologists recognize the role that reinforcement or reward plays in determining future behavior, but that this is not a “simple stamping-in process.” For beings as complex as humans, the effects of reinforcement depend upon an individual’s perception of a causal relationship between their behavior and the potential reward (Rotter & Hochreich, 1975).
A number of scales have been developed to measure locus of control (for an early review see Lefcourt, 1976), including one developed by Rotter himself (Rotter, 1966). Rotter’s scale, simply referred to as the I-E scale (for internal-external), consists of 29 forced-choice statements. For example:
1.a. Children get into trouble because their parents punish them too much.
1.b. The trouble with most children nowadays is that their parents are too easy with them.
In each instance, the person taking the test must choose one or the other option. After taking all 29, the person’s score is the total number of external choices. Does it seem difficult to determine whether 1.a. or 1.b. is the external choice? Good! Question 1 is actually a filler question, designed to interfere with the test taker’s ability to understand what the test is about! So, consider question 2:
2.a. Many of the unhappy things in people’s lives are partly due to bad luck.
2.b. People’s misfortunes result from the mistakes they make.
For question 2 it is quite obvious that choice a is the external choice, and if it wasn’t clear, the test has choice a marked for you!
Locus of control appears to arise from two primary sources: the family, and contingency awareness (Lefcourt, 1976). The role of the family in the development of locus of control is complex, and appears to be somewhat different based on the behavior of mothers and fathers. The most reliable finding appears to be that individuals with an internal locus of control had mothers who pushed them to achieve independence at an early age. This motherly push, however, must be a careful one. Children need support, guidance and nurturance, but they must not be smothered to the point of being pampered. Lefcourt (1976) cites Adler’s concern regarding two extremes in child-rearing, pampering and neglect, neither of which is conducive to the healthy psychological growth of a child. Contingency awareness refers to an understanding of instrumentality, the conception that one’s actions are indeed related to certain outcomes. In order for a child to repeat a behavior with purpose, the child must be able to recall that their prior actions resulted in a given outcome, and they must know that their actions were related to the expected outcomes. It would appear that children as young as two months old are capable of this type of social learning, and it tends to result in positive emotional reactions (Lefcourt, 1976).
Early studies on locus of control also focused on some interesting cultural questions. It is generally accepted that social class and ethnic group are important determinants of personality. Battle and Rotter (1963) found that lower class Blacks were significantly more external in their locus of control than were middle class Whites. Interestingly, middle class Blacks were closer to middle class Whites than lower class Whites were to middle class Whites, suggesting that social class may have been the primary factor in these results, rather than the race or ethnicity of the subjects. Furthermore, IQ seems to have exacerbated these results in that the most external individuals were high IQ lower class Blacks (i.e., individuals aware of social injustice in American society) and the most internal individuals were low IQ middle class Whites (who may be blaming themselves for failing to live up to their expected potential; Battle & Rotter, 1963). During the civil rights movement, Gore & Rotter (1963) examined whether locus of control might be a useful measure of social action. They found that students at a southern Black college who expressed interest in attending a civil rights rally or marching on the state capitol scored significantly more internal on the I-E scale. In other words, those who believed they could personally make a difference were more willing to try making that difference. In a study that followed soon after, Strickland (1965) compared Blacks who were indeed active in the civil rights movement to those who were not (but who were matched for sex, age, education, etc.). As predicted, the individuals who were active in the civil rights movement scored significantly more internal on the I-E scale than those who were not active. Strickland did note, however, that the individuals she studied were pioneers in the civil rights movement, and had become active, in part, because others groups had failed to demonstrate an adequate degree of commitment to the civil rights movement. Strickland’s concern seems to contradict earlier results of Mischel (1958a), who found that when individual’s make public commitments, they are less likely to change their expectancies (i.e., individuals publicly involved in the civil rights movement should have remained committed to the cause even when faced with initial failure). Still, as Mischel himself noted, one cannot rely entirely on inferences from research when considering the complexities of real-life (and, at the time, dangerous) behavior.
discussion question \(\PageIndex{2}\)
Do you consider yourself to have an internal or an external locus of control? Do you feel that locus of control is an important influence on personality; might it be good or bad?
Discussion Question: Do you consider yourself to have an internal or an external locus of control? Do you feel that locus of control is an important influence on personality; might it be good or bad?
Rotter’s Emphasis on Clinical Psychology
As noted above, Rotter was actively involved in developing the model that provided the basis for how clinical psychologists are typically trained today. Accordingly, much of Rotter’s career was devoted to clinical applications of his work, In addition to writing two books that emphasized clinical psychology (Rotter, 1954, 1964) and developing the I-E scale (Rotter, 1966), Rotter and one of his research assistants published The Rotter Incomplete Sentences Blank: College Form (Rotter and Rafferty, 1950). The book was intended to formalize the sentence completion method , particularly for use with college students. The test consists of forty simple statements that require the subject to finish the sentence. For example, one beginning is simply “My father…” The subjects responses are then scored in terms of whether they demonstrate conflict (on a scale of 1-3), are neutral, or whether they are positive (also on a scale of 1-3). The manual offers examples of possible answers for both males and females. For example, conflicted responses for males include breaking promises or being a fool (level 3), or never had much of a chance or is proud (level 1). A neutral response might simply be that the father is a salesman, or is a hard worker. Positive responses for females include that the father is quite a character or is a good man (level 1), or that he has a great sense of humor or is a lot of fun (level 3). Interpreting this test requires a great deal of experience, and an understanding of personality and human nature. Fortunately, Rotter and Rafferty include a number of individual cases as examples of how the Rotter Incomplete Sentences Blank can be used to evaluate individuals. Both the Rotter Incomplete Sentences Blank and the I-E scale have proven useful in evaluating patients, as well as normal individuals, in a variety of settings and cultures, including Africa, Sri Lanka, American Indians, Brazil, Black and White college students in America, Ukrainian doctors training in Canada, and amongst military personnel (Janzen, Paterson, Reid, & Everall, 1996; Lefcourt, 1976; Logan & Waehler, 2001; Nagelschmidt & Jakob, 1977; Niles, 1981; Picano, Roland, Rollins, & Williams, 2002; Rossier, Dahourou, & McCrae, 2005; Rotter, 1960, 1966; Trimble & Richardson, 1982). In a particularly interesting study, a unique version of the Sentence Completion Test was developed by Herbert Phillips and provided the basis for a major study on the personality of Thai peasants living in the village of Bang Chan, Thailand (Phillips, 1965). The Rotter Incomplete Sentences Blank, and other variations of the sentence completion method, remain very popular today (Holaday, Smith, & Sherry, 2000), ranking with the Rorschach Inkblot Test and the Thematic Apperception Test as the most popular projective tests for personality assessment.
Overall, Rotter emphasized the value of training clinical psychologists for just that responsibility, with a particular emphasis on the realities that will face the psychologist in an actual clinical setting (Rotter, 1954, 1964). In 1972, Rotter edited a volume including both original and previously published papers in which social learning theory was applied to psychopathology in general (Phares, 1972) and to such diverse topics as drinking amongst college students, excessively needy individuals, working with mentally retarded children, and electroconvulsive shock therapy (Cromwell, 1972; Dies, 1968; Jessor, Carman, & Grossman, 1968; Jessor, Liverant, & Opochinsky, 1963). A particularly important aspect of therapy also addressed in this volume is the issue of terminating therapy. Strickland & Crowne (1963) found that defensiveness and avoiding self-criticism are common signs in individuals who are likely to end therapy abruptly, whereas Piper, Wogan, & Getter (1972) found that the patient’s expectancy regarding improvement, and the value they place on improving, are useful predictors of terminating therapy. Although helping patient’s to achieve a level of psychological health that allows terminating therapy should be the goal of every therapist, premature termination might prove even more detrimental to the patient. For Rotter, the proper training of clinical psychologists is not an easy task. In the preface to Clinical Psychology , Rotter wrote:
…Yet psychology itself is a relatively new science and its areas of application are in rapid transition. Neither theory nor “facts” are always agreed upon, and in clinical psychology there is no single set of orthodox, approved skills for which a person can be certified as a trained practitioner…The goal is to gain comprehension without resorting to an oversimplification of the complex nature of man or of the problem of understanding him. (pg. xi; Rotter, 1964).
Behavioral Specificity and Consistency
In 1968, Walter Mischel challenged both state and trait theories of personality. Psychological states typically fall with the domain of psychodynamic theory, whereas trait theories are a perspective unto themselves. According to Mischel (1968), although state and trait theorists use very different language, they tend to approach personality in the same general way: they use responses to infer pervasive, underlying mental structures that exert enduring causal effects on behavior. Thus, both state and trait theorists emphasize consistency in behavior. However, there is a wealth of data that individuals do not act consistently from situation to situation. Instead, Mischel argues, behavior can best be predicted only when one takes into account the specific situation in which the behavior occurs:
Progress in the area of personality psychology and assessment has been hindered by the failure to apply relevant principles about the conditions that produce, maintain, and modify social behavior. The principles that emerge from basic research too often have not been seen as directly relevant to the understanding of the determinants of test responses in the clinic or the assessment project. It is as if we live in two independent worlds: The abstractions and artificial situations of the laboratory and the realities of life. (pg. 1; Mischel, 1968).
In order to support his argument, Mischel examined which aspects of behavior are or are not consistent. Generally, intellect is consistent, including academic ability, achievement, and cognitive style. In contrast, there is little evidence to support consistency of behavior across situations when examining personality variables such as attitudes, moral behavior, sexual identification, dependency, aggression, tolerance, conditionability, etc. (Mischel, 1968). How, then, might we predict behavior? Mischel suggests a dynamic perspective on how persons interact with their situations. If the environment has not changed much, we can expect past behavior to be a reasonable predictor of current behavior (and state and trait theories would seem to hold true as well). However, if the environment changes dramatically, the individual may act in unpredictable ways. In addition, the individual may begin to learn new social conditions, thus allowing for considerable change in behavior over time:
Global traits and states are excessively crude, gross units to encompass adequately the extraordinary complexity and subtlety of the discriminations that people constantly make. Traditional trait-state conceptions of man have depicted him as victimized by his infantile history, as possessed by unchanging rigid trait attributes, and as driven inexorably by unconscious irrational forces…A more adequate conceptualization must take full account of man’s extraordinary adaptiveness and capacities for discrimination, awareness, and self-regulation…and that an understanding of how humans can constructively modify their behavior in systematic ways is the core of a truly dynamic personality psychology. (pg. 301; Mischel, 1968)
Delayed Gratification
Perhaps Mischel’s most famous contribution to psychology is his research on delayed gratification. In a series of studies, begun in the late 1950s, Mischel examined the conditions under which children choose immediate gratification or whether they can delay gratification in order to obtain a larger reinforcer at a later time. The ability to delay gratification, according to Mischel, is essential for the development of self-control. From early childhood throughout the lifespan, achieving long-term goals often requires setting aside tempting distractions. Conversely, many personal and social problems result from failures of self-control, such as dropping out of school, poor job performance, and even violent and criminal behavior (Mischel & Mischel, 1980). In an amazing longitudinal study, Mischel and his colleagues offered 4 year-old children the opportunity to grab a marshmallow. But, if the child could wait until the researcher ran an errand, the child could then have two marshmallows! Some children grabbed the marshmallow as soon as the experimenter left, but others were able to wait 15-20 minutes. It was not easy, however. The children who waited demonstrated a variety of behaviors to distract themselves from the marshmallow: they would play, sing, cover their eyes so they didn’t have to look at the marshmallow, etc. The most striking results from this study were actually obtained years later. Mischel and his colleagues tracked down the former 4 year-old subjects as they were graduating from high school. The individuals who had delayed gratification as 4 year-olds were significantly more personally effective and self-assertive, and they were better able to cope with life’s frustrations (Mischel, Shoda, & Rodriguez, 1989; Shoda, Mischel, & Peake, 1990). In addition, the 4 year-old children who had been able to delay gratification were more successful as students in a variety of ways, including eventually earning significantly higher SAT scores (210 points higher, on the combined score), and the ability to delay gratification proved to be a better predictor of SAT scores than IQ (Peake, cited in Goleman, 1994).
Although the famous marshmallow-grabbing study was conducted at a preschool on the campus of Stanford University, Mischel began this research with very different groups: Black and East Indian children on the islands of Trinidad and Grenada (Mischel, 1958b, 1961). On these relatively poor, Caribbean islands, Mischel not only compared the Black and East Indian children, he also compared the children of Trinidad to the children of Grenada. The main purpose of the second study, however, was to examine the effect of fathers being absent from the home on the preference of children for immediate or delayed gratification. Overall, when fathers are absent from the home, both young boys and young girls (ages 8 to 9 years old) demonstrated a preference for immediate gratification. Mischel suggests that the inability to delay gratification amongst children who lack a father may be related to immaturity or poor psychological adjustment (Mischel, 1961).
While Mischel was at Stanford University, he also collaborated with Bandura. Blending the interests of both men, they examined whether observing models would affect children’s choices regarding immediate vs. delayed gratification. They identified two groups of children (both boys and girls) as preferring either immediate gratification (a small candy bar now) or delayed gratification (a larger candy bar later). The children were then exposed to either a live model choosing the alternative strategy, a symbolic model (a description of an adult choosing the alternative), or no model. As expected, exposure to a model choosing the alternative strategy dramatically affected the behavior of the children, and a live model was more effective than the symbolic model. The effects of this modeling appeared to be quite persistent (Bandura & Mischel, 1965). Considering the importance that modeling can play in developing the ability to delay gratification, it is perhaps easy to see why children in families lacking a complete and stable family structure don’t develop self-control as well as other children.
discussion question \(\PageIndex{3}\)
Mischel’s most famous contribution is the concept of delayed gratification. How good are you at waiting for gratification? Are some rewards easier to wait for than others? If you know anyone who is significantly different than you, either wanting immediate gratification or being able to delay it without much trouble, does the difference between you create any problems or interesting situations?
The Cognitive-Affective Processing System (CAPS)
More recently, Mischel has turned his attention to solving what has been called the “ personality paradox :” How do we reconcile our intuition and theories that personality is relatively stable with the overwhelming evidence that personality varies across different situations? Mischel proposes a dynamic personality system that takes into account both: (1) the behavioral consistency that accounts for specific scores on trait tests and indicates what the individual is like in general; and (2) the consistency in how an individual varies across different situations. This consistency of variation is recognized by distinct patterns of if…then … relationships, which are characteristic of the individual’s overall personality (Mischel, 2004).
In 1995, Mischel and Shoda first presented this dynamic approach to understanding personality, referring to it then as the cognitive-affective personality system, but now preferring the term cognitive-affective processing system (CAPS; Mischel, 2004; Mischel & Shoda, 1995/2000; Shoda, Leetiernan, & Mischel, 2002). Over a number of years, Mischel, his students, and his colleagues studied children extensively in a residential summer camp. They observed both behaviors and the situations in which they occurred. Over time, they were able to identify patterns of if…then… situation-behavior relations that reflected distinctive and stable characteristics of each child’s behavior organization. These observations, therefore, gave rise to situation-behavior profiles for each child. It is essential to recognize, however, that the term “situation” in these studies does not refer to simple environmental stimuli, as they might for a behaviorist such as B. F. Skinner. Instead, these situations activate a whole set of internal reactions, including cognitive and emotional elements. They are also not limited to the external world; they can be generated in thought, fantasy, planning, etc. Accordingly, Mischel and Shoda referred to these personality variables as cognitive-affective units (or CAUs). These CAUs include encodings, expectancies and beliefs, affects, goals and values, and competencies and self-regulatory plans.
Mischel and Shoda (1995/2000) did not neglect the individual’s development in this theory. Our ability to recognize distinct aspects of the environment are influenced by genetic/biological factors, cultural factors, and the interactions between them. These genetic/biological/cultural factors also influence the CAPS, as does our social learning history. In a sense, bringing all of these factors together begins to move us beyond the person-situation debate, since both sides of the debate are correct in the proper context. The future of personality theory may lie in an as yet undetermined synthesis of these perspectives (Fleeson, 2004). For now, according to Mischel, this dynamic approach to understanding personality has at least helped to bring together the major aspects of different schools of personality theory:
The two goals - dispositions and dynamics - that have so long been pursued separately do not require two fields from this perspective. In this theory, dispositions are conceptualized not in semantic terms but as processing structures characterized by stable cognitive affective organizations in the processing system that becomes activated when the individual encounters relevant situational features. (pg. 170; Mischel & Shoda, 1995/2000)


2.11: Power sets
Power set is a curious name for a simple concept. We talk about the power set “of" another set, which is the set of all subsets of that other set. Example: suppose \(A\) = { Dad, Lizzy }. Then the power set of \(A\) , which is written as “ \(\mathbb{P}(A)\) " is: { { Dad, Lizzy }, { Dad }, { Lizzy }, \(\varnothing\) }. Take a good look at all those curly braces, and don’t lose any. There are four elements to the power set of \(A\) , each of which is one of the possible subsets. It might seem strange to talk about “ all of the possible subsets" — when I first learned this stuff, I remember thinking at first that there would be no limit to the number of subsets you could make from a set. But of course there is. To create a subset, you can either include, or exclude, each one of the original set’s members. In \(A\) ’s case, you can either (1) include both Dad and Lizzy, or (2) include Dad but not Lizzy, or (3) include Lizzy but not Dad, or (4) exclude both, in which case your subset is \(\varnothing\) . Therefore, \(\mathbb{P}(A)\) includes all four of those subsets.
Now what’s the cardinality of \(\mathbb{P}(X)\) for some set \(X\) ? That’s an interesting question, and one well worth pondering. The answer ripples through the heart of a lot of combinatorics and the binary number system, topics we’ll cover later. And the answer is right at our fingertips, if we just extrapolate from the previous example. To form a subset of \(X\) , we have a choice to either in clude, or else ex clude, each of its elements. So there’s two choices for the first element 1 , and then whether we choose to include or exclude that first element, there are two choices for the second. Regardless of what we choose for those first two, there are two choices for the third, etc. So if \(|X|=2\) (recall that this notation means “ \(X\) has two elements" or “ \(X\) has a cardinality of 2"), then its power set has \(2 \times 2\) members. If \(|X|=3\) , then its power set has \(2 \times 2 \times 2\) members. In general:
\[|\mathbb{P}(X)| = 2^{|X|}.\]
As a limiting case (and a brain-bender) notice that if \(X\) is the empty set, then \(\mathbb{P}(X)\) has one (not zero) members, because there is in fact one subset of the empty set: namely, the empty set itself. So \(|X|=0\) , and \(|\mathbb{P}(X)|=1\) . And that jives with the above formula.
- I know there’s really no “first” element, but work with me here.


13.5: End-of-Chapter Material
- Cancer starts with the change in one cell in the body.
- Cancer cells do not like each other and break off from the other cells to spread throughout the body, called Metastasis.
- A cancerous tumor is called malignant, a non-cancerous tumor is called benign.
- Cancer is the 2nd leading cause of death.
- Many cancers are asymptomatic.
- The acronym CAUTION is helpful for remembering some common cancer symptoms.
- Lifestyle choices can impact your risk of getting cancer.
Media Attributions
- Diagram showing how cancer cells keep on reproducing to form a tumour © Cancer Research UK is licensed under a CC BY-SA (Attribution ShareAlike) license
- Diagram showing a tumour forcing its way through normal tissues © Cancer Research UK is licensed under a CC BY-SA (Attribution ShareAlike) license
- Tumor_Types_MTK © WolfpackBME is licensed under a CC BY-SA (Attribution ShareAlike) license
- How_metastasis_occurs_illustration © Unknown Illustrator is licensed under a Public Domain license
- Cancer_warning_signs © Linda Bartlett (Photographer) is licensed under a Public Domain license
- Melanoma_vs_normal_mole_ABCD_rule_NCI_Visuals_Online © Unknown creator; six images merged by User:Stevenfruitsmaak is licensed under a Public Domain license


1.1: Reflection - Introduction to Occupational Safety and Health
It's Your Turn
New World of Work (NWOW) Assessment: What kind of worker are you? How do you view worker safety?
Self Awareness
As we begin our exploration of work it is only fitting that you reflect on your experiences and perspectives of and with work. One of your first activities was to introduce yourself and share some of your background. This next activity is one designed to have you get a better understanding of why you may have certain views on work and worker safety, being ever mindful of who you are.
The Keirsey assessment is one of many types of personality profiles that seek to place individuals in groups with specific traits that may align with career paths or competencies. It is therefore reasonable that personality traits may also shape how you will view and approach safety in the workplace.
Access the Keirsey Assessment here or the 16personalities assessment and record your results. If you use the 16personalities assessment you will need to contrast your results with slides 6 and 17 below to determine which quadrant you belong. You may also just speak to your attribute without associating with it with the Keirsey temperament.
Reflect on your assessment by choosing one of the self image attributes (slide 17) of the Keirsey Overview sharing how that attribute would make you a safer worker. Provide detail, minimum one paragraph.
Using the Four Keirsey Temperaments
|
Guardian ESTJ, ISTJ, ESFJ, ISFJ
|
Artisan ESFP, ISFP, ESTP, ISTP
|
|---|---|
|
Idealist ENFJ, INFJ, ENFP, INFP
|
Rational ENTJ, INTJ, ENTP, INTP
|
Seeing yourself in the Four Temperaments
| Artisan | Guardian | Idealist | Rational | |
|---|---|---|---|---|
| Self Confidence | Adaptability | Respectability | Authenticity | Willpower |
| Self Esteem | Action | Reliability | Empathy | Ingenuity |
| Self Respect | Audacity | Service | Benevolence | Autonomy |
NWOW-Do you value all workers?
Social Diversity
Respecting and valuing differences is a basic tenet for the future of work.  When there are differences in who matters, there will be challenges to keeping all workers safe. The following personal reflection activities are designed to encourage you to value differences to ensure all workers matter.
Gender and Ethnicity
Use the the following excerpts from the NWOW social diversity lesson 1 to complete the questionnaire and class discussion at the end of the lesson.
Attributes of Social Diversity Awareness
- Respectful of differences in others’ backgrounds and beliefs in local communities and the world at large.
- Values diversity in the workplace, including gender, sexual orientation, ethnicity, and age. Understands these differences can actually improve products, services, or work processes.
Understanding Sex and Gender
| Sex Characteristics | Gender Characteristics |
|---|---|
| Biological sex refers to the biological and physiological differences between men and women. | Gender combines several elements: chromosomes (X or Y), anatomy, hormone levels, psychology, and culture. |
| Best identified through DNA, the clearest division between a male human and a female human is the type of chromosomes they carry. | “Gendering” refers to the psycho-social division of labor in a society; not to the biological and physiological differences between men and women. |
| A female human carries two X chromosomes while a male human carries one X chromosome and one Y chromosome. (The Columbia Encyclopedia, 2008) | Gendering patterns begin early to develop particular skills, beliefs, and attitudes in young “men” vs. young “women” |
| But, there are cases where an XY embryo fails to develop male anatomy and is identified as female at birth, while an XX embryo can develop male anatomy and is identified as as male at birth. (National Geographic, 2016) | Gendering varies dramatically across cultures. The degree of gender differentiation in a country is highly dependent on its national culture. |
| Self-identifying one’s gender, and a rejection of the traditional him/her categories, can be both a social and political statement |
Gendering in the Workplace
- There are still inequalities that exist in modern work environments for the types of jobs considered more appropriate for men and women.
- Can you name some examples of types of work where you feel there is a stereotype of it being a “male” job or a “female” job?
- There are also inequalities in pay scales for men and women occupying the same positions.
- The 2017 Economic Justice Report showed a 20% wage gap between men and women in the United States. It estimated that based on the rate of pay from 1960-2015, women are not projected to reach pay equity until 2059.
- This pay gap increases for both women and men of color.
Understanding Race and Ethnicity
| Race | Ethnicity |
|---|---|
| Sorts people into ethnic groups according to perceived physical and behavioral human characteristics | Allows people to self-identify with groupings of people on the basis of presumed (and usually claimed) commonalities including language, history, nation or region of origin, customs, ways of being, religion, names, physical appearance, and/or genealogy or ancestry |
| Associates differential value, power, and privilege with these characteristics and establishes a social status ranking among the different groups | Can be a source of meaning, action, and identity |
| Emerges (a) when groups are perceived to pose a threat (political, economic, or cultural) to each other’s world view or way of life; and/or (b) to justify the denigration and exploitation (past, current, or future) of, and prejudice toward, other groups | Confers a sense of belonging, pride, and motivation |
| So, defining someone by “race” is not only an outdated category, but it is based upon a history of exclusion and prejudice | When asking about background, ask what a person self-identifies as their ethnicity, not their race |
Class Activity
Having open discussions and striving to understand others from their own perspective, not from yours, is the first step in Social/Diversity Awareness.
What is your global literacy quotient? Take the Quiz. The passing score averages 30%
Next answer the following questions:
- What is your self-identified ethnicity?
- Have you ever felt you were defined by gender and/or race, and what impact did this have on you?
- How do discussions of what makes us different help expand the idea of what is “normal”?
- Is it important to include physical/cognitive impairments and workforce generation in the conversation of what is "normal" in Social Diversity?
Remember, you want to avoid situations like this:


2.5: Zeta Potential Analysis
Introduction
The physical properties of colloids (nanoparticles) and suspensions are strongly dependent on the nature and extent of the particle-liquid interface. The behavior of aqueous dispersions between particles and liquid is especially sensitive to the ionic and electrical structure of the interface.
Zeta potential is a parameter that measures the electrochemical equilibrium at the particle-liquid interface. It measures the magnitude of electrostatic repulsion/attraction between particles and thus, it has become one of the fundamental parameters known to affect stability of colloidal particles. It should be noted that that term stability, when applied to colloidal dispersions, generally means the resistance to change of the dispersion with time. Figure \(\PageIndex{1}\) illustrates the basic concept of zeta potential.
From the fundamental theory’s perspective, zeta potential is the electrical potential in the interfacial double layer (DL) at the location of the slipping plane (shown in Figure \(\PageIndex{1}\) ). We can regard zeta potential as the potential difference between the dispersion medium and the stationary layer of the fluid attached to the particle layer. Therefore, in experimental concerns, zeta potential is key factor in processes such as the preparation of colloidal dispersions, utilization of colloidal phenomena and the destruction of unwanted colloidal dispersions. Moreover, zeta potential analysis and measurements nowadays have a lot of real-world applications. In the field of biomedical research, zeta potential measurement, in contrast to chemical methods of analysis which can disrupt the organism, has the particular merit of providing information referring to the outermost regions of an organism. It is also largely utilized in water purification and treatment. Zeta potential analysis has established optimum coagulation conditions for removal of particulate matter and organic dyestuffs from aqueous waste products.
Brief History and Development of Zeta Potential
Zeta potential is a scientific term for electrokinetic potential in colloidal dispersions. In prior literature, it is usually denoted using the Greek letter zeta, Ζ, hence it has obtained the name zeta potential as Ζ-potential. The earliest theory for calculating Zeta potential from experimental data was developed by Marian Smoluchowski in 1903 (Figure \(\PageIndex{2}\) ). Even till today, this theory is still the most well-known and widely used method for calculating zeta potential.
Interestingly, this theory was originally developed for electrophoresis. Later on, people started to apply his theory in calculation of zeta potential. The main reason that this theory is powerful is because of its universality and validity for dispersed particles of any shape and any concentration. However, there still some limitations to this early theory as it was mainly determined experimentally. The main limitations are that Smoluchowski’s theory neglects the contribution of surface conductivity and only works for particles which have sizes much larger than the interface layer, denoted as κ a (1/κ is called Debye length and a is the particle radius).
Overbeek and Booth as early pioneers in this direction started to develop more theoretical and rigorous electrokinetic theories that were able to incorporate surface conductivity for electrokinetic applications. Modern rigorous electrokinetic theories that are valid almost any κa mostly are generated from Ukrainian (Dukhin) and Australian (O’Brien) scientists.
Principle of Zeta Potential Analysis
Electrokinetic Phenomena
Because an electric double-layer (EDL) exists between a surface and solution, then any relative motion between the rigid and mobile parts of the EDL will result in the generation of an electrokinetic potential. As described above, zeta potential is essentially a electrokinetic potential which rises from electrokinetic phenomena. So it is important to understand different situations where electrokinetic potential can be produced. There are generally four fundamental ways which zeta potential can be produced, via electrophoresis, electro-osmosis, streaming potential, and sedimentation potential as shown from Figure \(\PageIndex{3}\).
Calculations of Zeta Potential
There are many different ways of calculating zeta potential . In this section, the methods of calculating zeta potential in electrophoresis and electroosmosis will be introduced.
Zeta Potential in Electrophoresis
Electrophoresis is the movement of charged colloidal particles or polyelectrolytes, immersed in a liquid, under the influence of an external electric field. In such case, the electrophoretic velocity, v e (ms -1 ) is the velocity during electrophoresis and the electrophoretic mobility, u e (m 2 V -1 s -1 ) is the magnitude of the velocity divided by the magnitude of the electric field strength. The mobility is counted positive if the particles move toward lower potential and negative in the opposite case. And therefore, we have the relationship v e = u e E , where E is the externally applied field.
Thus, the formula accounted for zeta potential in electrophoresis case is given in EQ, where ε rs is the relative permittivity of the electrolyte solution, ε 0 is the electric permittivity of vacuum and η is the viscosity.
\[ \mathit{u}_{e}\ =\frac{\varepsilon _{rs} \varepsilon_{0} \zeta}{\eta } \label{1} \]
\[ \mathit{v}_{e}\ =\frac{\varepsilon _{rs} \varepsilon_{0} \zeta}{\eta } E \label{2} \]
There are two cases regarding the size of κa:
- κa < 1: the formula is similar, \ref{3} .
- κa > 1: the formula is rather complicated and we need to solve equation for zeta potential, \ref{4} , where \( y^{e \zeta}=\ e\zeta /kT \), m is about 0.15 for aqueous solution.
\[ \mathit{u}_{e} = \frac{2}{3} \frac{\varepsilon _{rs} \varepsilon_{0} \zeta}{\eta } \label{3} \]
\[ \frac{3}{2}\frac{\eta e}{\varepsilon _{rs} \varepsilon _{0}kT} \mathit{u_{e}} =\frac{3}{2}y^{ek} -\frac{6[\frac{y^{ek}}{2}-\frac{ln\ 2}{\zeta}\{1-e^{-\zeta y^{ek}}\}]}{2+ \frac{ka}{1+3m/\zeta ^{2}}e^{\frac{-\zeta y^{ek}}{2}}} \label{4} \]
Zeta Potential in Electroosmosis
Electroosmosis is the motion of a liquid through an immobilized set of particles, a porous plug, a capillary, or a membrane, in response to an applied electric field. Similar to electrophoresis, it has the electroosmotic velocity, v eo (ms -1 ) as the uniform velocity of the liquid far from the charged interface. Usually, the measured quantity is the volume flow rate of liquid divided by electric field strength, Q eo ,E (m 4 V -1 s -1 ) or diveided by the electric current, Q eo ,I (m 3 C -1 ). Therefore, the relationship is given by \ref{5} .
\[ Q_{eo} =\ \int \int v_{eo} dS \label{5} \]
Thus the formula accounted for Zeta potential in electroosmosis is given in EQ.
As with electrophoresis there are two cases regarding the size of κa:
- κa >>1 and there is no surface conduction, where Ac is the cross-section area and KL is the bulk conductivity of particle.
- κa < 1, \ref{6} , where \(\Delta u \ =\frac{K^{\sigma }}{K_{L}} \) is the Dukhin number account for surface conductivity, \( K^{\sigma} \) is the surface conductivity of the particle.
\[ Q_{eo , E} =\frac{-\varepsilon _{rs} \varepsilon_{0} \zeta }{\eta} Ac \nonumber \]
\[ Q_{eo , I} =\frac{-\varepsilon _{rs} \varepsilon_{0} \zeta }{\eta} \frac{1}{K_{L}} \nonumber \]
\[ Q_{eo , I} =\frac{-\varepsilon _{rs} \varepsilon_{0} \zeta }{\eta} \frac{1}{K_{L}(1+2\Delta u)} \label{6} \]
Relationship Between Zeta Potential and Particle Stability in Electrophoresis
Using the above theoretical methods, we can calculate zeta potential for particles in electrophoresis. The following table summarizes the stability behavior of the colloid particles with respect to zeta potential. Thus, we can use zeta potential to predict the stability of colloidal particles in the electrokinetic phenomena of electrophoresis.
| Zeta Potential (mV) | Stability behavior of the particles |
| 0 to ±5 | Rapid Coagulation or Flocculation |
| ±10 to ±30 | Incipient Instability |
| ±30 to ±40 | Moderate Stability |
| ±40 to ±60 | Good Stability |
| More than ±61 | Excellent Stability |
Instrumentation
In this section, a market-available zeta potential analyzer will be used as an example of how experimentally zeta potential is analyzed. Figure \(\PageIndex{4}\) shows an example of a typical zeta potential analyzer for electrophoresis.
The inside measuring principle is described in the following diagram, which shows the detailed mechanism of zeta potential analyzer (Figure \(\PageIndex{5}\) ).
When a voltage is applied to the solution in which particles are dispersed, particles are attracted to the electrode of the opposite polarity, accompanied by the fixed layer and part of the diffuse double layer, or internal side of the "sliding surface". Using the following formula below of this specific Analyzer and the computer program, we can obtain the zeta potential for electrophoresis using this typical zeta potential analyzer (Figure \(\PageIndex{6}\).


13.10: Operating Activities
Learning Outcomes
- Identify cash flows that result from operating activities
You have been asked to figure out cash flow for your company. Luckily, your supervisor has only asked you to do the operating activities section! But what items are included here? As you look through the income statement and balance sheet, what stuff do you need to be looking for? Also, what the heck is coming in (inflows) and going out (outflows)? Let’s watch a few examples first and then tackle the operating activities portion of our cash flow statement.
Learn More
Visit this site for an example and explanation of cash flow from operating activities indirect method .
Learn More
Also note, there are two methods to compute a cash flow statement. Direct and indirect. Take a look at this video for an explanation of the two methods. Remember companies only use one of these methods, and it is typically the direct method, but you should be familiar with both!
There are several sections that comprise the cash flow statement. The first portion of the cash flow statement includes what are called operating activities. These are cash transactions that happen in the normal course of business, affecting the revenue and expense accounts on your income statement. Operating activities can include the following items:
| Description | Inflow | Outflow |
| Collect cash from your customers | X | |
| Pay for inventory | X | |
| Pay your bills! (utilities, rent, insurance) | X | |
| Pay your employees | X | |
| Pay interest on loans | X | |
| Pay your taxes | X |
Collecting cash if the only cash inflow here! The other items are all cash leaving your business, also called outflows. So from now on, money coming in will be called an inflow and money going out will be called an outflow .
This portion of the cash flow statement can help you to better understand the need to have an effective accounts receivable system! If you sell product, but can’t collect the cash in a timely fashion, it may become difficult to meet your bill payment deadlines. If you don’t pay your bills on time, vendors, your employees and the government (especially the government) might not be happy with you.
In our budgeting module, we put together a cash budget. A cash budget is an important component of the financial health of all companies. Having enough cash coming in from customers and clients to cover the cash going out to meet payment responsibilities is crucial to successfully running or managing a business.
Practice Questions


7.5: Independent Events
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Learning Objectives
In this section, you will:
- Define independent events
- Identify whether two events are independent or dependent
In the last section, we considered conditional probabilities. In some examples, the probability of an event changed when additional information was provided. This is not always the case. The additional information may or may not alter the probability of the event.
In Example \(\PageIndex{1}\) we revisit the discussion at the beginning of the previous section and then contrast that with Example \(\PageIndex{2}\).
Example \(\PageIndex{1}\)
A card is drawn from a deck. Find the following probabilities.
- The card is a king.
- The card is a king given that the card is a face card.
Solution
a. Clearly, \(P\)(The card is a king) = 4/52 = 1/13.
b. To find \(P\)(The card is a king | The card is a face card), we reason as follows:
There are 12 face cards in a deck of cards. There are 4 kings in a deck of cards.
\(P\)(The card is a king | The card is a face card) = 4/12 = 1/3.
The reader should observe that in the above example,
\(P\)(The card is a king | The card is a face card) \(\neq\) \(P\)(The card is a king)
In other words, the additional information, knowing that the card selected is a face card changed the probability of obtaining a king.
Example \(\PageIndex{2}\)
A card is drawn from a deck. Find the following probabilities.
- The card is a king.
- The card is a king given that a red card has shown.
Solution
a. Clearly, \(P\)(The card is a king) = 4/52 = 1/13.
b. To find \(P\)(The card is a king | A red card has shown), we reason as follows:
Since a red card has shown, there are only twenty six possibilities. Of the 26 red cards, there are two kings. Therefore,
\(P\)(The card is a king | A red card has shown) = 2/26 = 1/13.
The reader should observe that in the above example,
\(P\)(The card is a king | A red card has shown) = \(P\)(The card is a king)
In other words, the additional information, a red card has shown, did not affect the probability of obtaining a king.
Whenever the probability of an event \(E\) is not affected by the occurrence of another event \(F\), and vice versa, we say that the two events \(E\) and \(F\) are independent. This leads to the following definition.
Definition: Independent
Two Events \(E\) and \(F\) are independent if and only if at least one of the following two conditions is true.
- \(\mathbf{P}(\mathbf{E} | \mathbf{F})=\mathbf{P}(\mathbf{E})\) or
- \(\mathbf{P}(\mathbf{F} | \mathbf{E})=\mathbf{P}(\mathbf{F})\)
If the events are not independent, then they are dependent.
If one of these conditions is true, then both are true.
We can use the definition of independence to determine if two events are independent.
We can use that definition to develop another way to test whether two events are independent.
Recall the conditional probability formula:
\[\mathrm{P}(\mathrm{E} | \mathrm{F})=\frac{\mathrm{P}(\mathrm{E} \cap \mathrm{F})}{\mathrm{P}(\mathrm{F})} \nonumber \]
Multiplying both sides by \(\mathrm{P}(\mathrm{F})\), we get
\[\mathrm{P}(\mathrm{E} \cap \mathrm{F})=\mathrm{P}(\mathrm{E} | \mathrm{F}) \mathrm{P}(\mathrm{F}) \nonumber \]
Now if the two events are independent, then by definition
\[\mathrm{P}(\mathrm{E} | \mathrm{F})=\mathrm{P}(\mathrm{E}) \nonumber \]
Substituting, \(P(E \cap F)=P(E) P(F)\)
We state it formally as follows.
Test For Independence
Two events \(E\) and \(F\) are independent if and only if
\[\mathbf{P}(\mathbf{E} \cap \mathbf{F})=\mathbf{P}(\mathbf{E}) \mathbf{P}(\mathbf{F}) \nonumber \]
In the Examples \(\PageIndex{3}\) and \(\PageIndex{4}\), we’ll examine how to check for independence using both methods:
- Examine the probability of intersection of events to check whether \(P(E \cap F)=P(E) P(F)\)
- Examine conditional probabilities to check whether \(P(E | F)=P(E)\) or \(P(F|E)=P(F)\)
We need to use only one of these methods. Both methods, if used properly, will always give results that are consistent with each other.
Use the method that seems easier based on the information given in the problem.
Example \(\PageIndex{3}\)
The table below shows the distribution of color-blind people by gender.
| Male(M) | Female(F) | Total | |
| Color-Blind(C) | 6 | 1 | 7 |
| Not Color-Blind(N) | 46 | 47 | 93 |
| Total | 52 | 48 | 100 |
where \(M\) represents male, \(F\) represents female, \(C\) represents color-blind, and \(N\) represents not color-blind. Are the events color-blind and male independent?
Solution 1: According to the test for independence, \(C\) and \(M\) are independent if and only if \(\mathrm{P}(\mathrm{C} \cap \mathrm{M})=\mathrm{P}(\mathrm{C}) \mathrm{P}(\mathrm{M})\).
From the table: \(P(C)\) = 7/100, \(P(M)\) = 52/100 and \(P(C \cap M)\) = 6/100
So \(P(C) P(M)\) = (7/100)(52/100) = .0364
which is not equal to \(P(C \cap M)\) = 6/100 = .06
Therefore, the two events are not independent. We may say they are dependent.
Solution 2: \(C\) and \(M\) are independent if and only if \(P(C|M) = P(C)\).
From the total column \(P(C)\) = 7/100 = 0.07
From the male column \(P(C|M)\) = 6/52= 0.1154
Therefore \(P(C|M) \neq P(C)\), indicating that the two events are not independent.
Example \(\PageIndex{4}\)
In a city with two airports, 100 flights were surveyed. 20 of those flights departed late.
- 45 flights in the survey departed from airport A; 9 of those flights departed late.
- 55 flights in the survey departed from airport B; 11 flights departed late.
Are the events "depart from airport A" and "departed late" independent?
Solution 1
Let A be the event that a flight departs from airport A, and L the event that a flight departs late. We have
\(P(A \cap L)\) = 9/100, \(P(A)\) = 45/100 and \(P(L)\) = 20/100
In order for two events to be independent, we must have \(P(A \cap L) = P(A) P(L)\)
Since \(P(A \cap L)\) = 9/100 = 0.09
and \(P(A) P(L)\) = (45/100)(20/100) = 900/10000 = 0.09
the two events "departing from airport A" and "departing late" are independent.
Solution 2
The definition of independent events states that two events are independent if \(P(E|F)=P(E)\).
In this problem we are given that
\(P(L|A)\) = 9/45= 0.2 and \(P(L)\) = 20/100 = 0.2
\(P(L|A) = P(L)\), so events "departing from airport A" and "departing late" are independent.
Example \(\PageIndex{5}\)
A coin is tossed three times, and the events \(E\), \(F\) and \(G\) are defined as follows:
\(E\): The coin shows a head on the first toss.
\(F\): At least two heads appear.
\(G\): Heads appear in two successive tosses.
Determine whether the following events are independent.
- \(E\) and \(F\)
- \(F\) and \(G\)
- \(E\) and \(G\)
Solution
We list the sample space, the events, their intersections and the probabilities.
\begin{aligned}
&\mathrm{S}=\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTH}, \mathrm{HTT}, \mathrm{THH}, \mathrm{THT}, \mathrm{TTH}, \mathrm{TTT}\\
&\begin{array}{ll}
\mathrm{E}=\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTH}, \mathrm{HTT}\}, & \mathrm{P}(\mathrm{E})=4 / 8 \text { or } 1 / 2 \\
\mathrm{F}=\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTH}, \mathrm{THH}\}, & \mathrm{P}(\mathrm{F})=4 / 8 \text { or } 1 / 2 \\
\mathrm{G}=\{\mathrm{HHT}, \mathrm{THH}\}, & \mathrm{P}(\mathrm{G})=2 / 8 \text { or } 1 / 4 \\
\mathrm{E} \cap \mathrm{F}=\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTH}\}, & \mathrm{P}(\mathrm{E} \cap \mathrm{F})=3 / 8 \\
\mathrm{F} \cap \mathrm{G}=\{\mathrm{HHT}, \mathrm{THH}\}, & \mathrm{P}(\mathrm{F} \cap \mathrm{G})=2 / 8 \text { or } 1 / 4 \\
\mathrm{E} \cap \mathrm{G}=\{\mathrm{HHT}\} & \mathrm{P}(\mathrm{E} \cap \mathrm{G})=1 / 8
\end{array}
\end{aligned}
a. \(E\) and \(F\) will be independent if and only if \(P(E \cap F) = P(E) P(F)\)
\(P(E \cap F) = 3/8\) and \(P(E) P(F) = 1/2 \cdot 1/2 = 1/4\).
Since 3/8 ≠ 1/4, we have \(P(E \cap F) \neq P(E) P(F)\).
Events \(E\) and \(F\) are not independent.
b. \(F\) and \(G\) will be independent if and only if \(P(F \cap G) = P(F) P(G)\).
\(P(F \cap G) = 1/4\) and \(P(F) P(G) = 1/2 \cdot 1/4 =1/8\).
Since 3/8 ≠ 1/4, we have \(P(F \cap G) \neq P(F) P(G)\).
Events \(F\) and \(G\) are not independent.
c. \(E\) and \(G\) will be independent if \(P(E \cap G) = P(E) P(G)\)
\(P(E \cap G) = 1/8\) and \(P(E) P(G) = 1/2 \cdot 1/4 =1/8\)
Events \(E\) and \(G\) are independent events because \(P(E \cap G) = P(E) P(G)\)
Example \(\PageIndex{6}\)
The probability that Jaime will visit his aunt in Baltimore this year is .30, and the probability that he will go river rafting on the Colorado river is .50. If the two events are independent, what is the probability that Jaime will do both?
Solution
Let \(A\) be the event that Jaime will visit his aunt this year, and \(R\) be the event that he will go river rafting.
We are given \(P(A)\) = .30 and \(P(R)\) = .50, and we want to find \(P(A \cap R)\).
Since we are told that the events \(A\) and \(R\) are independent,
\[P(A \cap R)=P(A) P(R)=(.30)(.50)=.15 \nonumber \]
Example \(\PageIndex{7}\)
Given \(P(B | A) = .4\). If A and B are independent, find \(P(B)\).
Solution
If \(A\) and \(B\) are independent, then by definition \(P(B | A) = P(B)\)
Therefore, \(P(B) = .4\)
Example \(\PageIndex{8}\)
Given \(P(A) =.7\), \(P(B| A) = .5\). Find \(P(A \cap B)\).
Solution 1
By definition \(P(B | A)=\frac{P(A \cap B)}{P(A)}\)
Substituting, we have
\[.5=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{.7} \nonumber \]
Therefore, \(P(A \cap B) = .35\)
Solution 2
Again, start with \(P(B | A)=\frac{P(A \cap B)}{P(A)}\)
Multiplying both sides by \(P(A)\) gives
\[P(A \cap B)=P(B | A) P(A)=(.5)(.7)=.35 \nonumber \]
Both solutions to Example \(\PageIndex{8}\) are actually the same, except that in Solution 2 we delayed substituting the values into the equation until after we solved the equation for \(P(A \cap B)\). That gives the following result:
Multiplication Rule for events that are NOT independent
If events \(E\) and \(F\) are not independent
\[\mathbf{P}(\mathbf{E} \cap \mathbf{F})=\mathbf{P}(\mathbf{E} | \mathbf{F}) \mathbf{P}(\mathbf{F}) \quad \text { and } \quad \mathbf{P}(\mathbf{E} \cap \mathbf{F})=\mathbf{P}(\mathbf{F} | \mathbf{E}) \mathbf{P}(\mathbf{E}) \nonumber \]
Example \(\PageIndex{9}\)
Given \(P(A) =.5\), \(P(A \cup B ) = .7\), if \(A\) and \(B\) are independent, find \(P(B)\).
Solution
The addition rule states that
\[\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \nonumber \]
Since \(A\) and \(B\) are independent, \(P(A \cap B)=P(A) P(B)\)
We substitute for \(P(A \cap B)\) in the addition formula and get
\[\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A}) \mathrm{P}(\mathrm{B}) \nonumber \]
By letting \(P(B) = x\), and substituting values, we get
\[\begin{array}{l}
.7=.5+x-.5 x \\
.7=.5+.5 x \\
.2=.5 x \\
.4=x
\end{array} \nonumber \]
Therefore, \(P(B) = .4\)


Skip to main content
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Communication is the act of transferring information from one person to another.
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True
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False
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Which of the following are forms of communication?
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Visual and written
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Verbal and non-verbal
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All of the above
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None of the above
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Interpersonal communication is the exchange of information, feelings, and meanings through verbal and non-verbal messages.
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True
-
False
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If you are a good talker, you are a good communicator.
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True
-
False
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Approximately what percentage of our day is spent communicating?
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25%
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50%
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75%
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90%
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What is the most widely used form of communication?
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Talking
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Writing
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Reading
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Listening
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Verbal communication is how we express ourselves in words and includes enunciation, pauses, and stutters.
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True
-
False
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The potential for miscommunication through short communication by email or text is insignificant.
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True
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False
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In what format do people best communicate and interpret tone?
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Texts
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Letters
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Emails
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Vocal messages
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Why is the KISS principle used?
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People are simple.
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Not everyone is able to understand complex communications.
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The more simply something is stated, the less opportunity there is for confusion.
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The more complexly something is stated, the less opportunity there is for confusion.
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Which of the following is considered non-verbal communication?
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Facial expressions and body language
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Personal space and personal appearance.
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All of the above
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None of the above
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Body language is the same regardless of where you come from or what your culture is.
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True
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False
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There are cultural differences in how people communicate and what is considered acceptable.
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True
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False
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Which of the following is not an example of open body language?
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Staying alert
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Sitting and facing the speaker
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Closing your eyes while an individual is speaking
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Nodding to acknowledge you’ve heard what was said
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A negative group environment can harmfully impact the motivation of co-workers and reduce productivity.
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True
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False
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Which of the following does not constitute effective communication?
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Being an active listener
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Accepting people’s differences
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Treating all people with respect
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Providing conditional acceptance
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Effective communication promotes understanding.
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True
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False
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In general, with whom is a more formal communication style used?
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Suppliers
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Friends and colleagues
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Employers, supervisors, and clients
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Colleagues, friends, suppliers, and clients
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An informal communication style is used more with employers than with friends.
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True
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False


5: Continuous Random Variables
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- 5.1: Introduction
- Continuous random variables have many applications. Baseball batting averages, IQ scores, the length of time a long distance telephone call lasts, the amount of money a person carries, the length of time a computer chip lasts, and SAT scores are just a few. The field of reliability depends on a variety of continuous random variables. In this chapter and the next, we will study the uniform distribution, the exponential distribution, and the normal distribution.
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- 5.2: Continuous Probability Functions
- The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to P(a<x<b)P(a<x<b) . The cumulative distribution function (cdf) gives the probability as an area.
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- 5.4: The Exponential Distribution
- The exponential distribution is often concerned with the amount of time until some specific event occurs. Values for an exponential random variable occur in the following way. There are fewer large values and more small values. The exponential distribution is widely used in the field of reliability. Reliability deals with the amount of time a product lasts.
Contributors
-
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at<EMAIL_ADDRESS>.