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?

Answer

N/a

Explanation

Let `b` represents the boy and `g` represents the girl and if a family has 2 children then sample space is
`S={"bb,bg,gb,gg"}`
`=n(S)=4`
Let `E=` event that both children are girls, then
`E={"gg"}impliesn(E)=1`
(i) Let `F=` event that smaller child is a girl, then
`F={"bg,gg"}impliesn(F)=2`
`impliesEnnF={"gg"}impliesn(EnnF)=1`
`:.P(E)=1/4, P(F)=2/4=1/2` and `(EnnF)=1/4`
Required probability
`=P(E/F)=(P(EnnF))/(P(F))=(1//4)/(2//4)=1/2`
(ii) Let `F=` event that at least one children is girl then
`F={"bg,gb,gg"}impliesEnnF={"gg"}=E`
`impliesn(F)=3,n(EnnF)=1`
`:.` Required probability
`=P(E/F)=(P(EnnF))/(P(F))`
`=(P(E))/(P(F))=(1//4)/(3//4)=1/3`