Question

The probability that $A$ speaks truth is $\frac{4}{5},$ while this probability for $B$ is $\frac{3}{5}.$ The probability of at least one of them is true when asked to speak on an event is

• A. $\frac{4}{25}$
• B. $\frac{2}{25}$
• C. $\frac{3}{25}$
• D. $\frac{23}{25}$

Answer 1

The correct option is D $\frac{23}{25}$

Probability that $A$ speaks truth $P\left(A\right)=\frac{4}{5}$
$\Rightarrow P\left(\overline{A}\right)=\frac{1}{5}$
Probability that $B$ speaks truth $P\left(B\right)=\frac{3}{5}$
$\Rightarrow P\left(\overline{B}\right)=\frac{2}{5}$

Probability that at least one of $A$ and $B$ is true is
$1-P\left(\text{both are false}\right)$
$=1-P\left(\overline{A}\right)P\left(\overline{B}\right)$
$=1-\frac{1}{5}\times \frac{2}{5}$
$=1-\frac{2}{25}=\frac{23}{25}$