Question

$A$ and $B$ are two independent witnesses in a case. The probability that $A$ will speak 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. $\frac{xy}{1-xy}$
• B. $\frac{xy}{1-x-y+2xy}$
• C. $\frac{2xy}{x-y}$
• D. $\frac{xy}{1-x-y}$

Answer 1

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

Let $K$ be the event that both of them agree, $L$ be the event that they both speak the truth and $M$ be the event that they both lie.
Probability that both $A$ and $B$ speak the truth = $P\left(L\right)=xy$
Probability that both $A$ and $B$ lie = $P\left(M\right)=(1-x)(1-y)=1-x-y+xy$
We need to find $P\left(\frac{L}{K}\right)$
$P\left(\frac{L}{K}\right)=\frac{P\left(L\right)P\left(\frac{K}{L}\right)}{P\left(L\right)P\left(\frac{K}{L}\right)+P\left(M\right)P\left(\frac{K}{M}\right)}=\frac{xy\times 1}{\left(xy\right)1+(1-x-y+xy)1}=\frac{xy}{1-x-y+2xy}$