Question

The following facts described the students who took my Bayesian statistics class in a recent year:

•35% of the students were statistics grad students.

•25% of the students were biostatistics grad students.

•40% of the students were undergraduates or grad students from other departments.

•60% of the statistics grad students were women.

•75% of the biostatistics grad students were women.

•40% of the students in other categories were women.

(a) You drew a student at random from my class list for that year. What is the probability that the student you drew is a woman?

(b) Suppose that the student you drew was a woman. What is the probability that she was not a statistics grad student and not a biostatistics grad student, given that she was a woman?

Answer 1

a)

Let S, B, O denote the event that the students were from statistics, biostatistics or other departments.

Let W denote the event that the student is woman. Then.

P(S) = 0.35

P(B) = 0.25

P(O) = 0.4

P(W | S) = 0.6

P(W | B) = 0.75

P(W | O) = 0.4

By law of total probability,

P(W) = P(S) P(W | S) + P(B) P(W | B) + P(O) P(W | O)

= 0.35 * 0.6 + 0.25 * 0.75 + 0.4 * 0.4

= 0.5575

The probability that a rtandom student you drew is a woman is 0.5575

b)

Probability that a student was not a statistics grad student and not a biostatistics grad student, given that she was a woman = Probability that a student was from other departments grad student given that she was a woman

= P(O | W)

= P(W | O) * P(O) / P(W) {By Bayes theorem}

= 0.4 * 0.4 / 0.5575

= 0.287