Problem 56 Solution

Bayes' Theorem:

Let events A₁,...,A_k form a partition of the space S such that Pr(A_j)>0 for j=1,...,k, and let B be any event such that Pr(B)>0. Then for i=1,...,k,

Pr(A_i|B) = Pr(A_i)*Pr(B|A_i) / Sum for j=1 to k of Pr(A_j)*Pr(B|A_j).

The source of this problem is the May 1997 issue of the Actuarial Review. Applying this theorm:

B=boy selected
A₁=boy added
A₂=girl added
g=number of girls before baby is added.

Pr(A₂|B) = (1/2)*(2/(3+g)) / [(1/2)*(2/(3+g)) + (1/2)*(3/(3+g))] = 0.4

Michael Shackleford, A.S.A.