Question

Mother, father and son line up at random for a family picture
E : son on one end, F : father in middle, Determine P(E | F).

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Solution

Let m, f and s denote respectively the mother, the father and the son.

therefore space space space space straight S space equals space open curly brackets straight m space straight f space straight s comma space straight m space straight s space straight f comma space space straight f space straight m space straight s comma space space straight f space straight s space straight m comma space straight s space straight m space straight f comma space straight s space straight f space straight m close curly brackets comma

Now E: 'son on one end' and F: 'father in the middle',
i.e. E =

open curly brackets straight m space straight f space straight s comma space straight f space straight m space straight s comma space straight s space straight m space straight f comma space straight s space straight f space straight m close curly brackets space and space straight F space equals space left curly bracket straight m space straight f space straight s comma space space straight s space straight f space straight m right curly bracket

therefore space space space straight E space intersection space straight F space equals space left curly bracket straight m space straight f space straight s comma space space straight s space straight f space straight m right curly bracket space equals space straight F therefore space space space space straight P left parenthesis straight E space left enclose straight F right parenthesis space equals space fraction numerator straight P left parenthesis straight E space intersection space straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals fraction numerator straight P left parenthesis straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals space 1.

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