Assume that each born child 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?

Let B denote boy and G denote girl.

Then, the sample space of the given experiment will be:

S = {GG, GB, BG, BB}

Let E be the event that ‘both are girls’.

⇒ E = {GG}

(i) Let F be the event that ‘the youngest is a girl’.

⇒ F = {GG, BG}

……….(I)

Now, E ∩ F = {GG}

……….(II)

Now, we know that

By definition of conditional probability,

[Using (I) and (II)]

(ii) Let H be the event that ‘at least one is a girl’.

⇒ H = {GG, GB, BG}

……….(III)

Now, E ∩ H = {GG}

……….(IV)

Now, we know that

By definition of conditional probability,

[Using (III) and (IV)]

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