A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of…
A person is known to speak false once out of 4 times. If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is
$\frac{1}{37}$
$\frac{1}{5}$
$\frac{12}{37}$
$\frac{25}{37}$
Solution
$\mathrm{E}_1 \rightarrow$ Person speak truth
$E_2 \rightarrow$ Person speak false
$\mathrm{F} \rightarrow$ Getting a king card
$\begin{aligned}
& P\left(E_1\right)=\frac{3}{4}, P\left(E_2\right)=\frac{1}{4}, P\left(\frac{F}{E_1}\right)=\frac{4}{52}=\frac{1}{13}, P\left(\frac{F}{E_2}\right)=\frac{12}{13} \\
& \therefore P\left(\frac{E_1}{F}\right)=\frac{P\left(\frac{F}{E_1}\right) \cdot P\left(E_1\right)}{P\left(\frac{F}{E_1}\right) \cdot P\left(E_1\right)+P\left(\frac{F}{E_2}\right) \cdot P\left(E_2\right)} \\
& =\frac{\frac{1}{13} \cdot \frac{3}{4}}{\frac{1}{13} \cdot \frac{3}{4}+\frac{12}{13} \cdot \frac{1}{4}}=\frac{\frac{3}{52}}{\frac{3}{52}+\frac{12}{52}}=\frac{3}{15}=\frac{1}{5}
\end{aligned}$