Joint Probability and Venn Diagram: An Example, Statistical Note 5

A Venn Diagram is an important tool to visualize the joint probability. I take an example from my statistical note 3 to apply the Venn Diagram. Table 1 presents data on the number of training participants by sex of participants and food habit.

Table 1: Food habit of training participants by sex

In the crosstab above, let me take the joint probability that a randomly selected participant is a woman who is a vegetarian also, denoted by P(W intersection V) or P(W∩V) is the product of P(W) and P(V/W). P(W) is calculated as 16 divided by 40, 0.40. P(V/W) is 12 divided by 16, 0.75. P(W∩V) is the product of 0.4 and 0.75, equal to 0.30. This probability value is equal to the first cross-sectional cell value (12) between women column and vegetarian row divided by the grant total value (40). Another way of calculating the P(W∩V) is the product of P(V) and P(W/V).

The same events and calculations are shown in diagram 1 also. A set or an event W that the participants in the training are women, with the corresponding probability P(W) is shown by the blue circle with the probability value.  Likewise, a set or an event V that the participants in the training are vegetarians, with the corresponding probability P(V) is shown by the yellow circle with the probability value. The area of overlap or an intersection between two circles is an event (W∩V) that a randomly selected participant is a woman who is a vegetarian also, is indicated by the blue line. The calculation of P(W∩V) is explained in the green box linked to that blue line.

Diagram 1: Joint event and probability