Discussion Forum



1)

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 on a certain statement. The probability that the statement is true is


A) $\frac{x-y}{x+y}$

B) $\frac{xy}{1+x+y+xy}$

C) $\frac{x-y}{1-x-y+2xy}$

D) $\frac{xy}{1-x-y+2xy}$

Answer:

Option D

Explanation:

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 → P(E1) = xy

E2 = A and B both tell a lie → P(E2) = (1-x)(1-y)

E = A and B agree in a certain statement

Clearly, P(E/E1) = 1 and P(E/E2) = 1 The required probability is P(E1/E )

Using Baye's theorem P(E1/E )

 = $\frac{P(E_{1})P(E/E_{1})}{P(E_{1})P(E/E_{1})+P(E_{2})P(E/E_{2})}$

= $\frac{xy.1}{xy.1+(1-x)(1-y).1}=\frac{xy}{1-x-y+2xy}$