Probability and Contingency Table: An Example, Statistical Note 3

I have taken an example from my statistical notes 1 and 2 to show the process of calculating the marginal, conditional and joint probabilities using the data presented in a contingency table, also known as the cross tabulation or crosstab. The cell values also are added to the crosstab as shown in table 1 below:

Table 1: Food habit of training participants by sex

This example has two discrete random variables or categorical variables each with two mutually exclusive categories of response. One categorical variable is the sex of training participants which has two categories of response: women (W) or men (M). Another categorical variable is the food habit which also has two mutually exclusive categories: vegetarian (V) and non-vegetarian (NV).

In the column total row, the simple or marginal probability of an independent event of women denoted by P(W) is 0.40, that is 40 percent of total training participants are women. This is calculated by dividing the column total or the marginal total of 16 women participants in the contingency table divided by the grand total of 40 participants. Likewise, the simple or marginal probability of an independent event of men denoted by P(M) is calculated at 0.60. Similar processes are followed in the row total column as well to calculate the simple or marginal probabilities of vegetarians denoted by P(V) equal to 0.45 and non-vegetarians denoted by P(NV) equal to 0.55 (table 2).

Table 2: Calculation of Marginal, Conditional and Joint Probabilities

The conditional probability of a vegetarian, a dependent event, given among the women represented by P(V/W) is 12 divided by 16, which is equal to 0.75, that is 75 percent women are vegetarians. Likewise, the conditional probabilities of P(NV/W), P(V/M) and P(NV/M) can be calculated following the same process. The conditional probabilities are also shown in table 2.

The joint probability that a randomly selected participant is a woman who is a vegetarian also, denoted by P(W intersection V) or P(WÇV) is the product of P(W) and P(V/W), the product of 0.4 and 0.75, equal to 0.30. This probability value is equal to the cross-sectional cell value between women column and vegetarian row divided by the grant total value in the contingency table, as shown in tables 2 and 3. The joint probabilities of P(WÇNV), P(MÇV) and P(MÇNV) can be calculated by using the same process.

Table 3: Cell values, Joint probabilities and cell values as percentage of grand total