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Probability

Question

Wired Faculty App

A bag contains 3 red and 7 black balls. Two balls are selected at random with 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 with replacement.

therefore space space space space straight P left parenthesis straight R subscript 1 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 3 over 10 space equals 9 over 100.

straight P left parenthesis straight R subscript 2 right parenthesis space equals space straight P left parenthesis apostrophe straight R subscript 1 space straight R subscript 2 apostrophe space 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

equals space 3 over 10 cross times 3 over 10 plus 7 over 10 cross times 3 over 10 equals space 9 over 100 plus 21 over 100 equals 30 over 100.

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 space intersection space straight R subscript 2 right parenthesis over denominator straight P left parenthesis straight R subscript 2 right parenthesis end fraction space equals fraction numerator begin display style 9 over 100 end style over denominator begin display style 30 over 100 end style end fraction space equals 3 over 10.

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