Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that i. the youngest is a girl, ii. at least one is a girl?

Explanation

Let b and g represent the boy and the girl child, respectively. If a family has two children, the sample space will be
`S={(b,b),(b,g),(g,b),(g,g)}`
Let A be the event that both children are girls. Therefore, A = `{(g,g)}`
(i) Let B be the evetn that the youngest child is a girl. Therefore,
`B={(b,g),(g,g)}`
`thereforeP(B)=2/4=1/2andP(AnnB)=1/4`
The conditional probability that both are girls, given that the youngest child is a girl, is given by P(A/B).
`thereforeP(A//B)=(P(AnnB))/(P(B))=(1//4)/(1//2)=1/2`
(ii) Let C be event that at least one child is a girl. Therefore,
`C={(b,g),(g,b),(g,g)}`
`impliesAnnC={g,g}`
`impliesP(C)=3/4andP(Ann C)=1/4`
The conditional probability that both are girls, given that at least one child is girl, is given by P(A/C). Therefore,
`P(A//C)=(P(AnnC))/(P(C))=(1//4)/(3//4)=1/3`