Question

$A$ speaks truth in $75$% cases and $B$ in $80$% of the cases. In what percentage of the cases are they likely to contradict each other, narrating the same incident?

• A. $5$%
• B. $15\%$
• C. $35\%$
• D. $45\%$

Answer 1

The correct option is C

$35\%$

$A$speaks truth $75\%$

That is,

$p\left(A\right)=\frac{3}{4}$

$P(A\text{'})=\frac{1}{4}$

Similarly,

$B$ speaks truth $80\%$

That is,

$p\left(B\right)=\frac{4}{5}$

$p(B\text{'})=\frac{1}{5}$

While contradicting the narration

$\begin{array}{l}Probability=P\left(A\right)P\left(\overline{B}\right)+P\left(\overline{A}\right)P\left(B\right)\end{array}$

On substituting, we get,

$$\begin{gathered} =\frac{3}{4}\times \frac{1}{5}+\frac{1}{4}\times \frac{4}{5} \\ =\frac{7}{20} \\ =\frac{7}{20}\times 100\% \\ =35\% \end{gathered}$$

Hence, the correct answer is option (c)