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Probability

Question
CBSEENMA12033776

A man is known to speak the truth 5 out of 6 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.

Solution
Let E1 denote the event that six occurs and E2 denote the event that it is not a six.
therefore space space space space straight P left parenthesis straight E subscript 1 right parenthesis space equals space 1 over 6 comma space space space space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 5 over 6
Let A denote the event that the man reports that it is six.
therefore space space space space space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis space equals space Probability space that space the space man space speaks space truth space equals space 5 over 6
space space space space space space space space space straight P left parenthesis straight A divided by straight E subscript 2 right parenthesis space equals space Probability space that space the space man space tells space straight a space lie space equals space 1 over 6
therefore space space space space space by space Baye apostrophe straight s space Law
space space space space space space space space space space space space space space straight P left parenthesis straight E subscript 1 divided by straight A right parenthesis space equals fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis. space space space straight P left parenthesis straight A divided by straight E subscript 1 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis. space space straight P left parenthesis straight A divided by 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 A divided by straight E subscript 2 right parenthesis end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator begin display style 1 over 6 end style cross times begin display style 5 over 6 end style over denominator begin display style 1 over 6 end style cross times begin display style 5 over 6 end style plus begin display style 5 over 6 end style cross times begin display style 1 over 6 end style end fraction equals space fraction numerator begin display style 5 over 6 end style cross times begin display style 1 over 6 end style over denominator 2 space open parentheses begin display style 5 over 6 end style cross times begin display style 1 over 6 end style close parentheses end fraction space equals 1 half

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