Question
$$A$$ 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. 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 :
$$\frac{{xy}}{{1 - x - y + 2xy}}$$
Solution :
$$A$$ and $$B$$ will agree in a certain statement if both speak truth or both tell a lie. We define following events
$${E_1} = A$$ and $$B$$ both speak truth
$$ \Rightarrow P\left( {{E_1}} \right) = xy$$
$${E_2} = A$$ and $$B$$ both tell a lie
$$ \Rightarrow P\left( {{E_2}} \right) = \left( {1 - x} \right)\left( {1 - y} \right)$$
$$E = A$$ and $$B$$ agree in a certain statement
Clearly, $$P\left( {\frac{E}{{{E_1}}}} \right) = 1{\text{ and }}P\left( {\frac{E}{{{E_2}}}} \right) = 1$$
The required probability is $$P\left( {\frac{{{E_1}}}{E}} \right)$$
Using Baye’s theorem
$$\eqalign{
& P\left( {\frac{{{E_1}}}{E}} \right) = \frac{{P\left( {{E_1}} \right)P\left( {\frac{E}{{{E_1}}}} \right)}}{{P\left( {{E_1}} \right)P\left( {\frac{E}{{{E_1}}}} \right) + P\left( {{E_2}} \right)P\left( {\frac{E}{{{E_2}}}} \right)}} \cr
& = \frac{{xy.1}}{{xy.1 + \left( {1 - x} \right)\left( {1 - y} \right).1}} \cr
& = \frac{{xy}}{{1 - x - y + 2xy}} \cr} $$