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Probability

Question

Wired Faculty App

A bag contains 3 red and 7 black balls. Two balls are selected at random without replacement. If the second selected is given to be red. what is the probability that the first selected is also red?

Solution

Let R₁, R₂ be the events that red ball is drawn in the first and second draw. Here balls are drawn without replacement.

therefore space space space straight P left parenthesis straight R subscript 1 space space intersection space straight R subscript 2 right parenthesis space equals space straight P left parenthesis both space red space balls right parenthesis space equals space 3 over 10 cross times 2 over 9 space equals 1 over 15

straight P left parenthesis straight R subscript 2 right parenthesis space equals space straight P left parenthesis apostrophe straight R subscript 1 straight R subscript 2 apostrophe space or space space apostrophe BR subscript 2 apostrophe right parenthesis space equals space straight P left parenthesis straight R subscript 1 space straight R subscript 2 right parenthesis space plus space straight P left parenthesis BR subscript 2 right parenthesis space space space space space space space space space space space equals space 3 over 10 cross times 2 over 9 plus 7 over 10 cross times 3 over 9 space equals space 6 over 90 plus 21 over 90 equals space 27 over 90 space equals 3 over 10

Required probability =

straight P left parenthesis straight R subscript 1 divided by straight R subscript 2 right parenthesis space equals space fraction numerator straight P left parenthesis straight R subscript 1 intersection straight R subscript 2 right parenthesis over denominator straight P left parenthesis straight R subscript 2 right parenthesis end fraction space equals space fraction numerator begin display style 1 over 15 end style over denominator begin display style 3 over 10 end style end fraction space equals 1 over 15 cross times 10 over 3 space equals 2 over 9.

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