Dear Student

A couple has two children,

Let boy be denoted by b & girl be denoted by g

So, S = {(b, b) ,(b, g),(g, b), (g, g)}

To find probability that both children are males,
if known that at least one of children is male

Let E : Both children are males
F : At least one child is male

To find P(E|F)

E : Both children are males

E = {(b, b)}

P(E) = 1/4

F : At least one child is male

F = {(b, g), (g, b), (b, b)}

P(F) = 3/4

E ∩ F = {(b, b)}

P(E ∩ F ) = 1/4

P(E|F) = (𝑃(𝐸 ∩ 𝐹))/(𝑃(𝐹))

= (1/4)/(3/4)

= 1/3

∴ Required Probability is 𝟏/𝟑

(ii) Find the probability that both children are females, if it is known that the elder child is a female.

S = {(b, b) ,(b, g),(g, b), (g, g)}

To find the probability that both children are females, if from that the elder child is a female.

Let E : both children are females
F : elder child is a female

To find P(E|F)

E : both children are females

E = {(g, g)}

P(E) = 1/4

F : elder child is a female

F = {(g, b), (g, g)}

P(F) = 2/4=1/2

Also, E ∩ F = {(g, g)}

So, P(E ∩ F) = 1/4

P(E|F) = (𝑃(𝐸 ∩ 𝐹))/(𝑃(𝐹))

= (1/4)/(1/2)

= ½

Thanks