Question

In a room there are three students, each of which is equally likely to be male or female
independently of the others. Define the events:
A = {all the students are of the same sex},
B = {there is at most one female},
C = {the group of three students includes a male and a female}.

(a) Find sets A, B, and C then determine the event probabilities

(b) Find P(A \ B), P(B − A), and P(A|B).

Are A and C independent? Explain.

Answer 1

a) P(A) = P(all are male) + P(all are female)
= (1/2)(1/2(1/2) + (1/2)(1/2(1/2) = 1/8 + 1/8 = 1/4 = 0.25
P(B) = P(1 female) or P(0 female)
= (3C1) *(1/2) * (1/2)^2 + (1/2)^3
= 4/8 = 0.5
P(C) = P(1 male and 2 female) + P(2 male and 1 female)
= = (1/2)(1/2(1/2) + (1/2)(1/2(1/2) = 2/8 =0.25
b) P(A and B) = P(all are female ) = (1/2)(1/2(1/2) = 1/8
P(B-A) = P(B) - P(A and B) = 0.5 - (1/8) = 0.375

P(A/B) = P(A and B) / P(B) = (1/8) / .25 = 0.03125

P(A and C) = 0
therefore, A and C are not indepedent