Question

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

• A. $\frac{4}{5}$
• B. $\frac{1}{5}$
• C. $\frac{7}{20}$
• D. $\frac{3}{20}$

Answer 1

The correct option is C $\frac{7}{20}$

Consider the following events
$E=A$ speaks truth
and $F=B$ speaks truth
We have
$P\left(E\right)=4/5,P\left(F\right)=3/4$
$\therefore$ Required probability
$=P(E\cap \overline{F})\cup (\overline{E}\cap F)$
$=P(E\cap \overline{F})\cup (\overline{E}\cap F)$
$P(E\cap \overline{F})+P(\overline{E}\cap F)=P\left(E\right)+P\left(\overline{F}\right)+P\left(\overline{E}\right)P\left(F\right)$
$=4/5\times (1-3/4)+(1-4/5)\times 3/4=7/20$