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

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

Answer 1

The correct option is C

$\frac{7}{20}$

The explanation for the correct option

The probability that $A$ speaks truth is, $P\left(A\right)=\frac{4}{5}$.

The probability that $A$ lies is, $P\left(A\text{'}\right)=1-P\left(A\right)=1-\frac{4}{5}=\frac{1}{5}$.

The probability that $B$ speaks truth is, $P\left(B\right)=\frac{3}{4}$.

The probability that $B$ lies is, $P\left(B\text{'}\right)=1-P\left(B\right)=1-\frac{3}{4}=\frac{1}{4}$.

Thus, the probability that $A$ and $B$ contradict each other can be given by, $P\left(A\right)·P\left(B\text{'}\right)+P\left(A\text{'}\right)·P\left(B\right)=\left(\frac{4}{5}\times \frac{1}{4}\right)+\left(\frac{1}{5}\times \frac{3}{4}\right)=\frac{4}{20}+\frac{3}{20}=\frac{7}{20}$.

Therefore, the probability that they contradict each other when asked to speak on a fact is $\frac{7}{20}$.