Calculating the probability that a randomly selected person is a woman who is a vegetarian also: An Example, Statistical Note 1

Of the total participants in a training, 40% are women and of the total women participants 75% are vegetarian. A participant is selected at random who is a woman, what is the probability that she is a vegetarian?

Calculating the probability of a "AND" compound event that a randomly selected participant is a woman who is a vegetarian includes calculating the probabilities of other events. Several concepts are introduced while answering this question.

Let W be a simple event that a participant is a woman and the simple or the marginal probability of W represented by P(W) is 0.40, that is 40 percent of total participants are women.

Let V/W be a simple event that a participant is a vegetarian among the women participants. The conditional probability of vegetarians among women participants symbolized by P(V/W) is 0.75, that is 75 percent of women participants are vegetarians. Here the occurrence of the event of vegetarian women participants is dependent on the event of occurrence of women participants.

Let (W intersection V) or (W and V) is a "AND" compound event that a participant is a woman and a vegetarian. Here, the multiplication rule of two dependent events is applied. The joint probability of two dependent events is the product of a marginal probability and the conditional probability. In this case, the joint probability of an woman participant who is a vegetarian indicated by P(W intersection V) or P(W and V) in which both events of women participants and vegetarian women among all women participants occur is the product of P(W) and P(V/W) and that is equal to 0.40 multiplied by 0.75, equal to 0.30. It means that 30 percent of the total participants are women who are vegetarians.