A and B are two independent witnesses (i.e. there is no collusion 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. The probability that the statement is true is

A and B will agree in a certain statement if both speak  truth or both tell a lie. We define following events E1=A and B both speak truth ⇒PE1=xyE2=A and B both tell a lie ⇒PE2=(1−x)(1−y)E=A and B agree in a certain statement  Clearly, PE/E1=1 and PE/E2=1 The required probability is PE1/E. theoremPE1/E=PE1PE/E1PE1PE/E1+PE2PE/E2                =xy.1xy.1+(1−x)(1−y)⋅1                =xy1−x−y+2xy