Question

A couple has two children, (i) find the probability that both children are males, if it is known that at least one of the children is male. (ii) find the probability that both children are females, if ti is known that the elder child is a female.

Answer 1

Toolbox:P(BA)=P(BA)=(P(A∩B)P(A))(P(A∩B)P(A))

Sample space: S = {MM, MF, FM, FF}, where M = male and F = female.

(i) To find P (both children are males, if it is known that at least one of the children is male).

A: Event that both children are male, and B: event that at least one of them is a male.

A: {MM} and B: {MF, FM, MM} →→ P (A ∩∩ B) = {MM}

Probability that both are males, if we know one is a female = P(AB)=P(AB)=(P(A∩B)P(B))(P(A∩B)P(B))

Given S = {MM, MF, FM, FF}, we can see that: P (A) = 1414; P (B) = 3434; P (A ∩∩ B) = 1414

Therefore, P(BA)=P(BA)=(P(A∩B)P(A))=1434=13(P(A∩B)P(A))=1434=13

(ii) To find the probability that both are females, if we know that the elder child is female.

Let A: event that both are females and B: event that elder one is a female child.

A: {FF} and B: {FF, FF} →→ P (A ∩∩ B) = {FF}.

P (both are females given the elder child is female) = P(AB)=P(AB)=(P(A∩B)P(B))(P(A∩B)P(B))

Given S = {MM, MF, FM, FF}, we can see that: P (A) = 1414; P (B) = 24=1224=12; P (A ∩∩ B) = 1414

Therefore, P(BA)=P(BA)=(P(A∩B)P(B))=1412=12