Statistical Independence
3.4 B IVARIATE P ROBABILITIES
In this section we introduce a class of problems that involve two distinct sets of events, which we label A1, A2, c, AH and B1, B2, c, BK. These problems have broad applica-tion in business and economics. They can be studied by constructing two-way tables that develop intuition for problem solutions. The events Ai and Bj are mutually exclusive and collectively exhaustive within their sets, but intersections 1Ai> Bj2 can occur between all events from the two sets. These intersections can be regarded as basic outcomes of a random experiment. Two sets of events, considered jointly in this way, are called bivariate, and the probabilities are called bivariate probabilities. It is possible to apply the methods of this section to trivariate and higher-level probabilities, but with added complexity.
We also consider situations where it is difficult to obtain desired conditional prob-abilities, but where alternative conditional probabilities are available. It may be difficult to obtain probabilities because the costs of enumeration are high or because some critical, ethical, or legal restriction prevents direct collection of probabilities.
Table 3.6 illustrates the outcomes of bivariate events labeled A1, A2, c, AH and B1, B2, c, BK. If probabilities can be attached to all intersections 1Ai> Bj2, then the whole probability structure of the random experiment is known, and other probabilities of inter-est can be calculated.
Table 3.6 Outcomes for Bivariate Events
B1 B2 c BK
A1 P1A1> B12 P1A1> B22 c P1A1> BK2 A2 P1A2> B12 P1A2> B22 c P1A2> BK2
. . . . .
. . . . .
. . . . .
AH P1AH> B12 P1AH> B22 c P1AH> BK2
As a discussion example, consider a potential advertiser who wants to know both income and other relevant characteristics of the audience for a particular television show.
Families may be categorized, using Ai, as to whether they regularly, occasionally, or never
3.4 Bivariate Probabilities 123 watch a particular series. In addition, they can be categorized, using Bj, according to low-, middle-, and high-income subgroups. Then the nine possible cross-classifications can be set out in the form of Table 3.7, with H = 3 and K = 3. The subsetting of the pop-ulation can also be displayed using a tree diagram, as shown in Figure 3.8. Beginning at the left, we have the entire population of families. This population is separated into three branches, depending on their television-viewing frequency. In turn, each of these branches is separated into three subbranches, according to the family income level. As a result, there are nine subbranches corresponding to all combinations of viewing frequency and income level.
Table 3.7 Probabilities for Television Viewing and Income Example
Viewing Frequency High Income Middle Income Low Income Totals
Regular 0.04 0.13 0.04 0.21
Occasional 0.10 0.11 0.06 0.27
Never 0.13 0.17 0.22 0.52
Totals 0.27 0.41 0.32 1.00
Figure 3.8 Tree Diagram for Television Viewing and Income Example
High income Middle income Low income
High income Middle income Low income
High income Middle income Low income Regularly watch series
Occasionally watch series Never watch series
Whole population
Joint and Marginal Probabilities
In the context of bivariate probabilities the intersection probabilities,
P1Ai> Bj2, are called joint probabilities. The probabilities for individual events, P1Ai2 or P1Bj2, are called marginal probabilities. Marginal probabilities are at the margin of a table such as Table 3.7 and can be computed by summing the corresponding row or column.
Now it is necessary to obtain the probabilities for each of the event intersections. These probabilities, as obtained from viewer surveys, are all presented in Table 3.7. For example, 10% of the families have high incomes and occasionally watch series. These probabili-ties are developed using the relative frequency concept of probability, assuming that the survey is sufficiently large so that proportions can be approximated as probabilities. On this basis, the probability that a family chosen at random from the population has a high income and occasionally watches the show is 0.10.
124 Chapter 3 Elements of Chance: Probability Methods
To obtain the marginal probabilities for an event, we merely sum the corresponding mutually exclusive joint probabilities:
P1Ai2 = P1Ai> B12 + P1Ai> B22 + g + P1Ai> BK2
Note that this would be equivalent to summing the probabilities for a particular row in Table 3.7. An analogous argument shows that the probabilities for Bj are the column totals.
Continuing with the example, define the television-watching subgroups as A1, “reg-ular”; A2, “occasional”; and A3, “never.” Similarly define the income subgroups as B1,
“high”; B2, “middle”; and B3, “low.” Then the probability that a family is an occasional viewer is as follows:
P1A22 = P1A2> B12 + P1A2> B22 + P1A2> B32 = 0.10 + 0.11 + 0.06 = 0.27
Similarly, we can add the other rows in Table 3.7 to obtain P1A12 = 0.21 and P1A32 = 0.52.
We can also add the columns in Table 3.7 to obtain
P1B12 = 0.27 P1B22 = 0.41 and P1B32 = 0.32
Marginal probabilities can also be obtained from tree diagrams like Figure 3.9, which has the same branches as Figure 3.8. The right-hand side contains all of the joint probabilities, and the marginal probabilities for the three viewing-frequency events are entered on the main branches by adding the probabilities on the corresponding sub-branches. The tree-branch model is particularly useful when there are more than two events of interest. In this case, for example, the advertiser might also be interested in the age of the head of household or the number of children. The marginal probabilities for the various events sum to 1 because those events are mutually exclusive and mutually exhaustive.
Figure 3.9 Tree Diagram for the Television Viewing–
Income Example, Showing Joint and Marginal Probabilities
P(A1>B1) = .04
P(A2) = .27 P(A1) = .2
1
P(A3) = . 52
P(A1>B2) = .13
P(A1>B3) = .04
P(A3>B1) = .13
P(A3>B2) = .17
P(A3>B3) = .22 P(A2>B1) = .10
P(S) = 1 P(A2>B2) = .11
P(A2>B3) = .06
A1: Regularly watch A2: Occasionally watch A3: Never watch B1: High income B2: Middle income B3: Low income S : Sample space
In many applications we find that the conditional probabilities are of more interest than the marginal probabilities. An advertiser may be more concerned about the prob-ability that a high-income family is watching than the probprob-ability of any family watching.
The conditional probability can be obtained easily from the table because we have all the joint probabilities and the marginal probabilities. For example, the probability of a high-income family regularly watching the show is as follows:
P1A1u B12 = P1A1> B12 P1B12 = 0.04
0.27 = 0.15
3.4 Bivariate Probabilities 125 Table 3.8 shows the probability of the viewer groups conditional on income levels.
Note that the conditional probabilities with respect to a particular income group always add up to 1, as seen for the three columns in Table 3.8. This will always be the case, as seen by the following:
a
H
i=1P1Aiu Bj2 = aH
i=1
P1Ai> Bj2
P1Bj2 = P1Bj2 P1Bj2 = 1
The conditional probabilities for the income groups, given viewing frequencies, can also be computed, as shown in Table 3.9, using the definition for conditional probability and the joint and marginal probabilities.
To obtain the conditional probabilities of income given viewing frequency in Table 3.7, we divide each of the joint probabilities in a row by the marginal probability in the right-hand column. For example,
P1low income u occasional viewer2 = 0.06 0.27 = 0.22
Table 3.8 Conditional Probabilities of Viewing Frequencies, Given Income Levels Viewing Frequency High Income Middle Income Low Income
Regular 0.15 0.32 0.12
Occasional 0.37 0.27 0.19
Never 0.48 0.41 0.69
Table 3.9 Conditional Probabilities of Income Levels, Given Viewing Frequencies Viewing Frequency High Income Middle Income Low Income
Regular 0.19 0.62 0.19
Occasional 0.37 0.41 0.22
Never 0.25 0.33 0.42
We can also check, by using a two-way table, whether or not paired events are statis-tically independent. Recall that events Ai and Bj are independent if and only if their joint probability is the product of their marginal probabilities—that is, if
P1Ai> Bj2 = P1Ai2P1Bj2
In Table 3.7 joint events A2 (“occasionally watch”) and B1 (“high income”) have a prob-ability of
P1A2> B12 = 0.10 and
P1A22 = 0.27 P1B12 = 0.27
The product of these marginal probabilities is 0.0729 and, thus, not equal to the joint prob-ability of 0.10. Hence, events A2 and B1 are not statistically independent.
Independent Events
Let A and B be a pair of events, each broken into mutually exclusive and col-lectively exhaustive event categories denoted by labels A1, A2, . . . , AH and B1, B2, . . . , BK. If every event Ai is statistically independent of every event Bj, then A and B are independent events.
126 Chapter 3 Elements of Chance: Probability Methods
Since A2 and B1 are not statistically independent, it follows that the events “viewing frequency” and “income” are not independent.
In many practical applications the joint probabilities will not be known precisely. A sample from a population is obtained, and estimates of the joint probabilities are made from the sample data. We want to know, based on this sample evidence, if these events are independent of one another. We will develop a procedure for conducting such a test later in the book.