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Two independent witnesses, A and B, give the same statement. The probability that A tells the truth is x, and the probability that B tells the truth is y. If both agree on a statement, then the probability that the statement is true is

1. (x - y)/(x + y)
2. xy/(1 + x + y + xy)
3. (x - y)/(1 - x - y + 2xy)
4. xy/(1 - x - y + 2xy)

Correct answer: xy/(1 - x - y + 2xy)

Solution

P(true and agree)=xy, P(false and agree)=(1-x)(1-y). P(true|agree)=xy/[xy+(1-x)(1-y)] = xy/(1 - x - y + 2xy).

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