The probability that Classroom #2 was chosen?

Question

One of 2 small classrooms is chosen at random with equally likely probability, and then a student is chosen at random from the chosen classroom. Classroom #1 has 5 boys and 11 girls. Classroom #2 has 15 boys and 9 girls.

What is the probability that Classroom #2 was chosen at random, given that a girl was chosen?

Answer 1

By definition, the probability of A given B is P(A∩B)/P(B). So in this case, if we let C2 denote the event that classroom 2 was chosen and G be the event that a girl was chosen:

P(C2|G) = P(C2∩G)/P(G)

Now, P(C2∩G) is easy to compute: it is simply P(C2)*P(G|C2), which is 1/2 * 9/(9+15) = 3/16. P(G) is a little harder, but is clearly P((G∩C1) ∪ (G∩C2)) which, since these are disjoint events, is P(G∩C1) + P(G∩C2). P(G∩C2) has already been computed, and P(G∩C1) = P(C1)*P(G|C1) = 1/2 * 11/(11+5) = 11/32. Sp P(G) = 11/32 + 3/16 = 17/32. Plugging this into our formula:

P(C2|G) = P(C2∩G)/P(G) = (3/16)/(17/32) = 6/17