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Probability

Question

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A speaks truth in 60% cases and B in 90% cases. In what percentage of cases are they likely to contradict each other in stating the same fact.

Solution

Let P(A), P(B) be the probability of A and B speaking the truths. Then

straight P left parenthesis straight A right parenthesis space equals space 60 over 100 equals space 3 over 5 comma space space space straight P left parenthesis straight B right parenthesis space equals 90 over 100 space equals 9 over 10

therefore space space straight P left parenthesis straight A with bar on top right parenthesis space equals space straight P left parenthesis straight A space tells space straight a space lie right parenthesis space equals space 1 space minus space straight P left parenthesis straight A right parenthesis space equals space 1 minus 3 over 5 equals space 2 over 5

straight P left parenthesis straight B with bar on top right parenthesis space equals space straight P left parenthesis straight B space tells space straight a space lie right parenthesis space equals space 1 space minus space straight P left parenthesis straight B right parenthesis space equals space 1 minus 9 over 10 space equals 1 over 10

Now P(A and B contradict) =

straight P left parenthesis straight A right parenthesis. space straight P left parenthesis straight B with bar on top right parenthesis space plus space straight P left parenthesis straight A with bar on top right parenthesis. space straight P left parenthesis straight B right parenthesis

=

3 over 5 cross times 1 over 10 plus 2 over 5 cross times 9 over 10 space equals 3 over 30 plus 18 over 50 space equals 21 over 50 space equals 42 percent sign

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