Question

Two groups are competing for the position on the Board of directors of a corporation: The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.

Solution

Let E₁, E₂, E be the events
E₁ : ‘First group wins’,
E₂ : ‘Second group wins’,
E : ‘New product is introduced’
$therefore space space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 0.6 space equals space 6 over 10 space space space and space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 0.4 space equals space 4 over 10$
$straight P left parenthesis straight E vertical line straight E subscript 1 right parenthesis space equals 0.7 space equals space 7 over 10 space and space straight P left parenthesis straight E space vertical line space straight E subscript 2 right parenthesis space equals space 0.3 space equals space 3 over 10$
Required probability = $straight P left parenthesis straight E subscript 2 space vertical line thin space straight E right parenthesis$
$equals space fraction numerator straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight E vertical line straight E subscript 2 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight E vertical line straight E subscript 1 right parenthesis space plus space straight P left parenthesis straight E subscript 2 right parenthesis space straight P left parenthesis straight E vertical line straight E subscript 2 right parenthesis end fraction space space left parenthesis By space Baye apostrophe straight s space Theorem right parenthesis equals space fraction numerator begin display style 4 over 10 end style cross times begin display style 3 over 10 end style over denominator begin display style 6 over 10 end style cross times begin display style 7 over 10 end style plus begin display style 4 over 10 end style cross times begin display style 3 over 10 end style end fraction equals space fraction numerator 12 over denominator 42 plus 12 end fraction equals space 12 over 54 space equals 2 over 9.$

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