Bivariate Probabilities

In this section we introduce a class of problem that involve two distinct sets of events, which label A1, A2,…..An and B1, B2,…………,Bk. These problems have broad application 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  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.

The following Table illustrates the outcomes of bivariate events labeled A1, A2,…..An and B1, B2,…………,Bk. If probabilities can be attached to all intersections  , then the whole probability structure of the random experiment is known, and other probabilities of interest can be calculated.

 

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 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 the above Table, with i = 3 and k = 3. The sub setting of the  population can also be displayed using a tree diagram, as shown in the following Figure. 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 sub-branches, according to all combinations of viewing frequency and income level. As a result, there are nine sub-branches corresponding to all combinations of viewing frequency and income level.  

Figure 1: Tree diagram for television viewing and income example

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 1. For example, 10% of the families have high incomes and occasionally watch the series. These probabilities 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.

Table 2: Probability 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