# A and B are two independent witnesses (i.e., there is no collision between them) in a case. The probability

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A and B are two independent witnesses (i.e., there is no collision between them) in a case. The probability that A will speak the truth is x and the probability that B will speak the truth is y. A and B agree in a certain statement. What is the probability that the statement is true?

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A and B will agree on a certain statement if both speak the truth or both tell a lie. Now, let us define the events as follows:

E1: A and B both speak the truth ⇒ P(E1) = xy

E2: A and B both tell a lie ⇒ P(E2) = (1 – x) (1 – y)

E: A and B agree on a certain statement.

Clearly, P(E/E1) = P(E/E2) = 1

Now, we are required to find P(E1/E).

P(E1/E) = $\frac{P(E_1).P(E/E_1)}{P(E_1).P(E/E_1)+P(E_2).P(E/E_2)}$ = $\frac{xy.1}{xy.1+(1-x)(1-y).1}$ = $\frac{xy}{1-x-y+2xy}$