Question

In many countries around the world, couples look to a son to take care of them in their old age. They are inclined to keep having children until they have a son. For this problem, imagine that a government considers permitting a couple to continue having children until they have a son. Assume that the birth of a male is as likely as the birth of a female.
(a) What’s the probability that a couple will have only one child?
(b) What’s the probability that a family will have two children?
(c) What would be the average number of children per family?

Answer 1

P(birth of male child) = P(birth of female child) = 0.5

a)

P(couple will have only 1 child) = P(the first child born is a male) = 0.5

b)

P(couple will have only 2 child) = P(the first child born is a female and the second child born is a male) = (0.5)(0.5) = 0.25

c)

Let X denote the number of child per family.

P(X=1) = 0.5 and P(X=2) = 0.25 {From parts (a) and (b)}

P(X=3) = P(first 2 children born are females and the 3rd child born is a male)

= (0.5)(0.5)(0.5) = 0.125

and so on....

Now, Average number of children per family = E(X) =

Let

So,

So,

Thus,

Hence, which is the average number of children per family.