Sponsor Area

Probability

Question

A and B are two events such that P(A) ≠ 0. Find P (B|A), if
(i) A is a subset of B (ii) A ∩ B = ϕ

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Solution

(i) Since A is a subset of B

therefore space space space space space straight A space intersection thin space straight B space equals space straight A

...(1)
Now

straight P left parenthesis straight B space left enclose straight A right parenthesis end enclose space equals space fraction numerator straight P left parenthesis straight B intersection straight A right parenthesis over denominator straight P left parenthesis straight A right parenthesis end fraction space equals space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator straight P left parenthesis straight A right parenthesis end fraction space equals space fraction numerator straight P left parenthesis straight A right parenthesis over denominator straight P left parenthesis straight A right parenthesis end fraction

open square brackets because space of space left parenthesis 1 right parenthesis close square brackets

therefore space space space space straight P left parenthesis straight B space left enclose straight A right parenthesis end enclose space equals space 1

(ii)

straight P left parenthesis straight B space left enclose straight A right parenthesis end enclose space equals space fraction numerator straight P thin space left parenthesis straight B intersection straight A right parenthesis over denominator straight P left parenthesis straight A right parenthesis end fraction space equals space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator straight P left parenthesis straight A right parenthesis end fraction

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