S KEWNESS
7. A baseball batter has a number of different outcomes—such as a hit, walk, strikeout, fly ball out, and more—each time he or she is at bat
In each of the random experiments listed, we can specify the possible outcomes, defined as basic outcomes. We do not know in advance which outcome will occur.
3.1 Random Experiment, Outcomes, and Events 95 Sample Space
The possible outcomes from a random experiment are called the basic out-comes, and the set of all basic outcomes is called the sample space. We use the symbol S to denote the sample space.
Example 3.1 Professional Baseball Batter (Sample Space)
What is the sample space for a professional baseball batter? A high-quality professional baseball player, when at bat, could have the listed outcomes occur that are shown in the sample space displayed in Table 3.1. The sample space consists of six basic out-comes. No two outcomes can occur together, and one of the seven must occur. The probabilities were obtained by examining baseball batters’ data.
Table 3.1 Outcomes for a Baseball Batter
SAMPLE SPACE, S PROBABILITY
O1 Safe hit 0.30
O2 Walk or hit by pitcher 0.10
O3 Strikeout 0.10
O4 Groundball out 0.30
O5 Fly ball out 0.18
O6 Reach base on an error 0.02
Example 3.2 Investment Outcomes (Sample Space)
An investor follows the Dow Jones Industrial index. What are the possible basic out-comes at the close of the trading day?
Solution The sample space for this experiment is as follows:
S = [{1. The index is higher than at yesterday’s close}, {2. The index is not higher than at yesterday’s close}]
One of these two outcomes must occur. They cannot occur simultaneously. Thus, these two outcomes constitute a sample space.
In many cases we are interested in some subset of the basic outcomes and not the individual outcomes. For example, we might be interested in whether the batter reached the base safely—that is, safe hit, walk, or reach base on an error. This subset of outcomes is defined as an event.
We must define the basic outcomes in such a way that no two outcomes can occur simultaneously. In addition, the random experiment must necessarily lead to the occur-rence of one of the basic outcomes.
96 Chapter 3 Elements of Chance: Probability Methods
It is possible that the intersection of two events is the empty set. In the hitter example, if we had defined an event C, “batter is out,” then the intersection of events A, “batter reaches base safely,” and C would be an empty set, so A and C are mutually exclusive.
Event
An event, E, is any subset of basic outcomes from the sample space. An event occurs if the random experiment results in one of its constituent basic outcomes. The null event represents the absence of a basic outcome and is denoted by [.
Intersection of Events
Let A and B be two events in the sample space S. Their intersection, denoted by A> B, is the set of all basic outcomes in S that belong to both A and B.
Hence, the intersection A> B occurs if and only if both A and B occur. We use the term joint probability of A and B to denote the probability of the intersec-tion of A and B.
More generally, given K events E1, E2, . . . , EK, their intersection, E1> E2> . . . > EK, is the set of all basic outcomes that belong to every Ei1i = 1, 2, . . . , K2.
Mutually Exclusive
If the events A and B have no common basic outcomes, they are called mutu-ally exclusive, and their intersection, A> B, is said to be the empty set, indi-cating that A> B has no members.
More generally, the K events E1, E2, . . . , EK are said to be mutually exclusive if every pair (Ei, Ej) is a pair of mutually exclusive events.
In the batter example Events A and C from above are mutually exclusive.
Figure 3.1 illustrates intersections using a Venn diagram. In part (a) of Figure 3.1, the rectangle S represents the sample space, and the two closed figures represent the events A and B. Basic outcomes belonging to A are within the circle labeled A, and basic outcomes belonging to B are in the corresponding B circle. The intersection of A and B, A> B, is indicated by the shaded area where the figures intersect. We see that a basic outcome is in A> B if and only if it is in both A and B. Thus, in the batter example outcomes, safe hit, O1, or reach base on an error, O6, belong to both events: “the batter reaches base safely” (Event A [O1, O2, O6]) and “the batter hits the ball” (Event B [O1, O4, O5, O6]). In Figure 3.1(b) the figures do not intersect, indicating that events A and B are mutually exclusive. For example, if a set of accounts is audited, the events “less than In some applications we are interested in the simultaneous occurrence of two or more events. In the batter example we might be interested in two events: “the batter reaches base safely” (Event A [O1, O2, O6]) and “the batter hits the ball” (Event B [O1, O4, O5, O6]).
One possibility is that specific outcomes in both events occur simultaneously. This will happen for outcomes that are contained in both events—that is, safe hit, O1, or reach base on an error, O6. This later set of outcomes is the intersection A> B[O1, O6]. Thus, in the batter example the outcomes, safe hit, O1, or reach base on an error, O6, belong to both of these two events: “the batter reaches base safely” (Event A [O1, O2, O6]) and “the batter hits the ball” (Event B [O1, O4, O5, O6]). Note that the probability of this intersection is 0.32 10.30 + 0.022.
3.1 Random Experiment, Outcomes, and Events 97 5% contain material errors” and “more than 10% contain material errors” are mutually exclusive.
Table 3.2 Intersection of and Mutually Exclusive Events
(a) Intersection of Events (b) Mutually Exclusive Events
B B B B
A A> B A - 1A > B2 A [ A
A B - 1A > B2 A> B A B A> B
Union
Let A and B be two events in the sample space, S. Their union, denoted by A< B, is the set of all basic outcomes in S that belong to at least one of these two events. Hence, the union A< B occurs if and only if either A or B or both occur.
More generally, given the K events E1, E2, . . . , EK, their union, E1 < E2 < . . . < EK, is the set of all basic outcomes belonging to at least one of these K events.
The Venn diagram in Figure 3.2 shows the union, from which it is clear that a basic outcome will be in A< B if and only if it is in either A or B or both.
A S
B A
S
A˘B B
(a) (b)
Figure 3.1 Venn Diagrams for the Intersection of Events A and B: (a) A> B is the Shaded Area; (b) A and B are Mutually Exclusive
Tables 3.2(a) and 3.2(b) can also be used to demonstrate the same conditions. The en-tire table represents S the sample space. Basic outcomes belonging to A are in the first row labeled A, and basic outcomes belonging to B are in the first column labeled B. The second row designates basic outcomes not in A as A, and outcomes not in B as B. The intersection of A and B, A> B, is indicated by the upper left table cell. A basic outcome is in A > B if and only if it is in both A and B. Thus, in the batter example—Table 3.2(a)—outcomes safe hit, O1, and reach base on an error, O6, belong to the two events: “the batter reaches base safely” (Event A [O1, O2, O6]) and “the batter hits the ball” (Event B [O1, O4, O5, O6]), the result shown in Figure 3.1(a). In Table 3.2(b) the figures do not intersect, indicating that events A and B are mutually exclusive, the same as Figure 3.1(b). When we consider several events jointly, another possibility of interest is that at least one of them will occur.
This will happen if the basic outcome of the random experiment belongs to at least one of the events. The set of basic outcomes belonging to at least one of the events is called their union. For the batter example the two events, “the batter reaches base safely” (Event A [O1, O2, O6]) and “the batter hits the ball” (Event B [O1, O4, O5, O6]), the events [O1, O2, O4, O5, O6] are included in at least one of the events. This is an example of the union of two events.
98 Chapter 3 Elements of Chance: Probability Methods
If the union of several events covers the entire sample space, S, we say that these events are collectively exhaustive. Since every basic outcome is in S, it follows that every outcome of the random experiment will be in at least one of these events. In the baseball example, the events “the batter gets on base” and “batter makes an out” are collectively exhaustive.
Figure 3.2 Venn Diagram for the Union of Events A and B
Collectively Exhaustive
Given the K events E1, E2, . . . , EK in the sample space, S, if
E1 hE2 h. . . h EK = S, these K events are said to be collectively exhaustive.
We can see that the set of all basic outcomes contained in a sample space is both mu-tually exclusive and collectively exhaustive. We have already noted that these outcomes are such that one must occur, but no more than one can simultaneously occur.
Next, let A be an event. Suppose that our interest is all of the basic outcomes not in-cluded in A.
Complement
Let A be an event in the sample space, S. The set of basic outcomes of a ran-dom experiment belonging to S but not to A is called the complement of A and is denoted by A.
Clearly, events A and A are mutually exclusive—no basic outcome can belong to both—and collectively exhaustive—every basic outcome must belong to one or the other.
Figure 3.3 shows the complement of A using a Venn diagram. We have now defined three important concepts—intersection, union, and complement—that will be important in our development of probability.
A S
B
Figure 3.3 Venn Diagram for the Complement of
Event A A A
S
3.1 Random Experiment, Outcomes, and Events 99
Example 3.3 Batter Performance Showing Unions, Intersections, and Complements
The following examples help to illustrate these concepts. When a batter is up, two events of interest are “the batter reaches base safely” (Event A [O1, O2, O6]) and “the batter hits the ball”(Event B [O1, O4, O5, O6]), using the definitions from Example 3.1.
1. The complements of these events are, respectively, “the batter does not reach base safely” 1A2 and “the batter does not hit the ball” 1B2
A = 3O3, O4, O54 B = 3O2, O34
2. The intersection of A and B is the event “batter reaches base safely as the result of hitting the ball,” and so,
A> B = 3O1, O64 (3.1)
3. The union is the event “the batter reaches base safely or the batter hits the ball,”
and so,
A< B = 3O1, O2, O4, O5, O64 (3.2)
4. Note that the events A3O1, O2, O64 and A3O3, O4, O54 are mutually exclusive since their intersection is the empty set and collectively exhaustive since their union is the sample space S, that is,
A< A = 3O1, O2, O3, O4, O5, O64
The same statements apply for B3O1, O4, O5, O64 and B [O2, O3].
Consider also the intersection of events A3O3, O4, O54 and B3O1, O4, O5, O64. The events O4, “ground ball out,” and O5, “fly ball out,” represent the condition where the batter hits the ball but makes an out.
Example 3.4 Dow Jones Industrial Average (Unions, Intersections, and Complements)
We designate four basic outcomes for the Dow Jones Industrial average over two con-secutive days:
O1: The Dow Jones average rises on both days.
O2: The Dow Jones average rises on the first day but does not rise on the second day.
O3: The Dow Jones average does not rise on the first day but rises on the second day.
O4: The Dow Jones average does not rise on either day.
Clearly, one of these outcomes must occur, but more than one cannot occur at the same time. We can, therefore, write the sample space as S = 3O1, O2, O3, O44. Now, we consider these two events:
A: “The Dow Jones average rises on the first day.”
B: “The Dow Jones average rises on the second day.”
Find the intersection, union, and complement of A and B.
Solution We see that A occurs if either O1 or O2 occurs, and B occurs if either O1 or O3 occurs; thus,
A = 3O1, O24 and B = 3O1, O34
100 Chapter 3 Elements of Chance: Probability Methods
The intersection of A and B is the event “the Dow Jones average rises on the first day and rises on the second day.” This is the set of all basic outcomes belonging to both A and B, A> B = 3O14.
The union of A and B is the event “the Dow Jones average rises on at least one of the two days.” This is the set of all outcomes belonging to either A or B or both. Thus,
A< B = 3O1, O2, O34
Finally, the complement of A is the event “the Dow Jones average does not rise on the first day.” This is the set of all basic outcomes in the sample space, S, that do not belong to A. Hence,
A3O3, O44 and, similarly, B3O2, O44
Figure 3.4 shows the intersection of events A and B. This intersection contains all outcomes that belong in both A and B. Clearly, A> B = 3O34.
Figure 3.4 Venn Diagram for the Intersection of A and B
Additional results are shown in the chapter appendix.
E
XERCISESBasic Exercises
For Exercises 3.1–3.4 use the sample space S defined as follows:
S= 3E1, E2, E3, E4, E5, E6, E7, E8, E9, E104 3.1 Given A = 3E1, E3, E6, E94, define A.
3.2 Given A = 3E1, E3, E7, E94 and B = 3E2, E3, E8, E94.
a. What is A intersection B?
b. What is the union of A and B?
c. Is the union of A and B collectively exhaustive?
3.3 Given A = 3E1, E3, E7, E94 and B = 3E2, E3, E8, E94.
a. What is the intersection of A intersection B?
b. What is the union of A and B?
c. Is the union of A and B collectively exhaustive?
3.4 Given A = 3E3, E5, E6, E104 and B = 3E3, E4, E6, E94 a. What is the intersection of A and B?
b. What is the union of A and B?
c. Is the union of A and B collectively exhaustive?
Application Exercises
3.5 A corporation takes delivery of some new machinery that must be installed and checked before it becomes
available to use. The corporation is sure that it will take no more than 7 days for this installation and check to take place. Let A be the event “it will be more than 4 days before the machinery becomes available”
and B be the event “it will be less than 6 days before the machinery becomes available.”
a. Describe the event that is the complement of event A.
b. Describe the event that is the intersection of events A and B.
c. Describe the event that is the union of events A and B.
d. Are events A and B mutually exclusive?
e. Are events A and B collectively exhaustive?
f. Show that 1A > B2 < 1A > B2 = B.
g. Show that A< 1A > B2 = A < B.
3.6 Consider Example 3.4, with the following four basic outcomes for the Dow Jones Industrial Average over two consecutive days:
O1: The Dow Jones average rises on both days.
O2: The Dow Jones average rises on the first day but does not rise on the second day.
A S
B
A˘B
3.2 Probability and Its Postulates 101 O3: The Dow Jones average does not rise on the
first day but rises on the second day.
O4: The Dow Jones average does not rise on either day.
Let events A and B be the following:
A: The Dow Jones average rises on the first day.
B: The Dow Jones average rises on the second day.
a. Show that 1A > B2 < 1A > B2 = B.
b. Show that A< 1A > B2 = A < B.
3.7 Florin Frenti operates a small, used car lot that has three Mercedes (M1, M2, M3) and two Toyotas (T1, T2).
Two customers, Cezara and Anda, come to his lot,
and each selects a car. The customers do not know each other, and there is no communication between them. Let the events A and B be defined as follows:
A: The customers select at least one Toyota.
B: The customers select two cars of the same model.
a. Identify all pairs of cars in the sample space.
b. Define event A.
c. Define event B.
d. Define the complement of A.
e. Show that 1A > B2 < 1A > B2 = B.
f. Show that A< 1A > B2 = A < B.
3.2 P
ROBABILITY ANDI
TSP
OSTULATESNow, we are ready to use the language and concepts developed in the previous section to determine how to obtain an actual probability for a process of interest. Suppose that a random experiment is to be carried out and we want to determine the probability that a particular event will occur. Probability is measured over the range from 0 to 1. A prob-ability of 0 indicates that the event will not occur, and a probprob-ability of 1 indicates that the event is certain to occur. Neither of these extremes is typical in applied problems. Thus, we are interested in assigning probabilities between 0 and 1 to uncertain events. To do this, we need to utilize any information that might be available. For example, if incomes are high, then sales of luxury automobiles will occur more often. An experienced sales manager may be able to establish a probability that future sales will exceed the company’s profitability goal based on past experience. In this section we consider three definitions of probability:
1. Classical probability
2. Relative frequency probability 3. Subjective probability
