Example
Suppose there is a mixed school having 60% boys and 40% girls as students. The girl students wear trousers or skirts in equal numbers; the boys all wear trousers. An observer sees a (random) student from a distance; all the observer can see is that this student is wearing trousers. What is the probability this student is a girl? The correct answer can be computed using Bayes' theorem.
The event G is that the student observed is a girl, and the event T is that the student observed is wearing trousers. To compute P(G|T), we first need to know:
- P(G), or the probability that the student is a girl regardless of any other information. Since the observer sees a random student, meaning that all students have the same probability of being observed, and the percentage of girls among the students is 40%, this probability equals 0.4.
- P(G'), or the probability that the student is not a girl (i.e., a boy) regardless of any other information (G' is the complementary event to G). This is 60%, or 0.6.
- P(T|G), or the probability of the student wearing trousers given that the student is a girl. As they are as likely to wear skirts as trousers, this is 0.5.
- P(T|G'), or the probability of the student wearing trousers given that the student is a boy. This is given as 1.
- P(T), or the probability of a (randomly selected) student wearing trousers regardless of any other information. Since P(T) = P(T|G)P(G) + P(T|B)P(B) (via the law of total probability), this is 0.5×0.4 + 1×0.6 = 0.8.
Given all this information, the probability of the observer having spotted a girl given that the observed student is wearing trousers can be computed by substituting these values in the formula:
Read more about this topic: Posterior Probability
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