Joint and Marginal Probabilities

In the context of bivariate probabilities the intersection probabilities,  are called joint probabilities. The probabilities for individual events, P(Ai) or P(Bj), are called marginal probabilities. Marginal probabilities are at the margin of a table and can be computed by summing the corresponding row or column.
To obtain the marginal probabilities for an event, we merely sum the corresponding mutually exclusive joint probabilities:

Continuing with the example, define the television watching subgroups as A1, “Regular”; 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

Similarly, we can add the other rows in Table above to obtain P(A1) = 0.21 and P(A3) = 0.52.

We can also add the columns in Table to obtain

             P(B1) = 0.27                     P(B2) = 0.41                   P(B3) = 0.32

Marginal probabilities can also be obtained from tree diagrams such as Figure 2, which has the same branches as Figure 1. 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 marginal probabilities for the various events sum to 1 because those events are mutually exclusive and mutually exhaustive.

 

Figure 2: Tree diagram for the television viewing-income example, showing
               joint and marginal probabilities.                                                      

In many applications we find that the conditional probabilities are of more interest than a the marginal probabilities. An advertiser may be more concerned about the probability that a high-income family is watching than the probability 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

Table 3 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. This will always be the case, as seen by the following:

The conditional probabilities for the income groups, given viewing frequencies, can also be computed , as shown in Table 4, using the definition for conditional probability and the joint and marginal probabilities.

To obtain the conditional probabilities with respect to income groups in Table 2, we divide each of the joint probabilities in a row by the marginal probability in the right-hand column.

Table 3: 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
Totals	1.0	1.0	1.0

Table 4: Conditional Probabilities of Income Levels, Given the Viewing Frequencies.

Viewing Frequency	High Income	Middle Income	Low Income	Totals
Regular	0.19	0.62	0.19	1.0
Occasional	0.37	0.41	0.22	1.0
Never	0.25	0.33	0.42	1.0

We can also check, by using a two-way table, whether or not paired events are statistically independent. Events A and B are independent if and only if their joint probability is the product of their marginal probabilities—that is, if

Odds

Odds are used to communicate probability information in some situations. For example, a sports analyst might report that the odds in favor of team A winning over team B are 2 to 1. Odds can be converted directly to probabilities, and probabilities can be converted to odds using the following equation.

The odds in favor of a particular event are given by the ratio of the probability of the event divided by the probability of its complement. The odds in favor of A are

Similarly, if the odds in favor of winning are 3 to 2, then the probability of winning is 0.60. Note that 0.60/0.40 is equal to 3/2.