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 at least one is a girl?

Solution

Let the first child be denoted by capital letter and the second i.e. younger one by a smaller letter.
∴ S = {Bb, Bg, Gb, Gg}
Let E : both children are girls
∴ E = {Gg}
Let F : at least one is a girl
$therefore space space space space space space space space straight F space equals space open curly brackets Bg comma space Gb comma space Gg close curly brackets therefore space space space space straight E space intersection space straight F space equals space open curly brackets Gg close curly brackets therefore space space space straight P left parenthesis straight F right parenthesis space equals space 3 over 4 comma space space space straight P left parenthesis straight E intersection straight F right parenthesis space equals space 1 fourth$
Required probability = $straight P left parenthesis straight E space left enclose straight F right parenthesis space equals space space fraction numerator straight P left parenthesis straight E intersection straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals fraction numerator begin display style 1 fourth end style over denominator begin display style 3 over 4 end style end fraction space equals 1 third$

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Probability

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 at least one is a girl?

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