A family has two children Let C be a characteristic that a c

A family has two children. Let C be a characteristic that a child

can have, and assume that each child has characteristic C with probability p, inde-

pendently of each other and of gender. For example, C could be the characteristic

“born in winter”. Show that the probability that both children

are girls given that at least one is a girl with characteristic C is 2p , which is 1/3 4p

if p = 1 and approaches 1/2 from below as p 0.

Solution

The language of problem does not seem to be clear here. So I will take some assumptions and calculate the probability of both children being girls if it is given that atleast one child is a girl with characteristic C.

We assume the probability of a child to be a boy /girl to be 0.5.

The probability of a child to have characteristic C is p.

P(atleast one child is a girl with characteristic C.) =P(first child is a girl with characteristic C)+P(second child is a girl with characteristic C)-P(both child are girls with characteristic C)

P(atleast one child is a girl with characteristic C.) = (1/2)(p)+(1/2)(p)-(1/2)(p)(1/2)(p)

P(atleast one child is a girl with characteristic C.) = 3p/4

P(both children are girls and atleast one has characteristic C) = P(both children are girls)*P(atleast one has C)

P(both children are girls and atleast one has characteristic C) =(1/4)*(1-(1-p)²)

Let P\' denote the required probability .

P\'=P(both children are girls and atleast one has characteristic C) /P(atleast one child is a girl with characteristic C.)

P\' = (1-(1-p)²)/3p