Question

A family decides to have children until it has 4 children of the same gender. Assuming P(B) = P(G) = 0.5,

a) what is the pmf of X = the number of children in the family?

Answer 1

Given that P(B) = P(G) = 0.5

Let X denote the number of children in the family, and X takes values 3, 4 and 5.

If X =3,

P(Having 3 boys) = 0.5 * 0.5 * 0.5 = 0.5^3

P(having 3 Girls) = 0.5^3

P(X = 3) = 0.5^3 + 0.5^3 = 0.25

If X = 4,

P(E) = (0.5^3 * 0.5 + 0.5^3 * 0.5 + 0.5^3 * 0.5 + 0.5^3 * 0.5 + 0.5^3 * 0.5 + 0.5^3 * 0.5 +)

=0.375

If X = 5

P(E) = (0.5^3 + 0.5^3 + 0.5^3 + 0.5^3 + 0.5^3 + 0.5^3 + 0.5^3 + 0.5^3 )

=0.375

Probability mass function of X

X	0	1	2	3	4	5	6
P(X)	0	0	0	0.25	0.375	0.375	0