Question

$A$ 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. Then the probability that this statement is true is:

• A. $\frac{xy}{1-x-y+2xy}$
• B. $\frac{xy}{1-x-y+xy}$
• C. $\frac{xy}{1-2x-y+xy}$
• D. $\frac{xy}{1+x+2y-xy}$

Answer 1

The correct option is A $\frac{xy}{1-x-y+2xy}$

Let $E_1$ be the event both $A$ and $B$ speak the truth and $E_2$ be the event that both A and B will tell a lie.
Let $E$ be the event that $A$ and $B$ agree i.e., both $A,B$ speak the truth or both false.
$P\left(E_1\right)=x\times y,P\left(E_2\right)=(1-x)(1-y)$
Clearly,$P\left(\frac{E}{E_1}\right)=1,P\left(\frac{E}{E_2}\right)=1$
$P\left(E_1/E\right)=\left[\frac{P\left(E_1\right).P\left(E/E_1\right)}{P\left(E_1\right).P\left(E/E_1\right)+P\left(E_2\right).P\left(E/E_2\right)}\right]=\left[\frac{xy\times 1}{xy\times 1+(1-x)(1-y)\times 1}\right]=\left[\frac{xy}{1-x-y+2xy}\right]$