Question

A speaks truth in 75% of the cases, while B in 90% of the cases. In what percent of cases are they likely to contradict each other in stating the same fact?

Answer 1

Let

$A=$ Event that A speaks the truth

$B=$ Event that B speaks the truth

Then,

$P\left(A\right)=\frac{75}{100}=\frac{3}{4}$

$P\left(B\right)=\frac{90}{100}=\frac{9}{10}$

$P(A-$lie$)=$ $1-\frac{3}{4}=\frac{1}{4}$

$P(B-$lie$)=$ $1-\frac{9}{10}=\frac{1}{10}$

Now,

$A$ and $B$ contradict each other $=[A$ lies and $B$ true$]$ or $[B$ true and $B$ lies$]$

$=\text{P(A).P(B-lie) + P(A-lie).P(B)}$

$=\frac{3}{4}\times \frac{1}{10}+\frac{1}{4}\times \frac{9}{10}=\frac{12}{40}=\frac{3}{10}$

$=\frac{3}{10}\times 100$%$=30$%