Question

The probability that $A$ speaks truth is $\frac{4}{5}$ while this probability for $B$ is $\frac{3}{4}$. Find the probability that they contradict each other when asked to speak on a fact.

Answer 1

Let $E$ be the event that $B$ speaks truth and $F$ be the event that $A$ speaks truth.

Now $P\left(E\right)=\frac{3}{4}$ and $P\left(F\right)=\frac{4}{5}$

$\therefore P\left(AandBcontradicteachother\right)$

$=P\left[\left(BtellstruthandAtellslie\right)or\left(BtellslieandAtellstruth\right)\right]$

$=P[(E\cap \overline{F})\cup (\overline{E}\cap F)]$

$=P\left(E\right)P\left(\overline{F}\right)+P\left(\overline{E}\right)P\left(F\right)$

$=\frac{3}{4}\times \frac{1}{5}+\frac{1}{4}\times \frac{4}{5}$

$=\frac{3+4}{20}=\frac{7}{20}$