Question

P speaks truth in 70% of the cases and Q in 30% of the cases. In what per cent of cases are they likely to agree in stating the same fact?
Do you think, when they agree, means both are speaking truth?

Answer 1

Let $A=$ Event that A speaks the truth

$B=$ Event that B speaks the truth

Then, $P\left(A\right)=\frac{70}{100}=\frac{7}{10}$

$P\left(B\right)=\frac{30}{100}=\frac{3}{10}$

$P(A-lie$) $=$ $1-\frac{7}{10}=\frac{3}{10}$

$P(B-lie$) $=$ $1-\frac{3}{10}=\frac{7}{10}$

Now,

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

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

$=\frac{7}{10}\times \frac{7}{10}+\frac{3}{10}\times \frac{3}{10}=\frac{29}{20}$

$=\frac{29}{50}\times 100=58\%$