$A$ speaks truth in $75 \%$ of the cases and $B$ in $80 \%$ of the cases. Then, the probability that their…
$A$ speaks truth in $75 \%$ of the cases and $B$ in $80 \%$ of the cases. Then, the probability that their statements about an incident do not match, is
$\frac{7}{20}$
$\frac{3}{20}$
$\frac{2}{7}$
$\frac{5}{7}$
Solution
Let Event A: A speaks the truth
Event B: B speaks the truth
$\begin{aligned}
& P(A)=75 \%=\frac{75}{100}=\frac{3}{4} \\
& \text { and } P(B)=80 \%=\frac{80}{100}=\frac{4}{5}
\end{aligned}$
$\therefore \quad$ Required probability
$\begin{aligned}
& P(A \bar{B})+P(\bar{A} B)=P(A) \times P(\bar{B})+P(\bar{A}) \times P(B) \\
& =P(A) \times[1-P(B)]+P(A) \times P(B) \\
& =\frac{3}{4} \times\left(1-\frac{4}{5}\right)+\left(1-\frac{3}{4}\right) \times \frac{4}{5} \\
& =\frac{3}{4} \times \frac{1}{5}+\frac{1}{4} \times \frac{4}{5}=\frac{7}{20}
\end{aligned}$