A man speaks truth \( 2 \) out of \( 3 \) times. He picks one of the natural numbers in the set \( S=\{1,2, \),
\( 3,4,5,6,7\} \) and reports that it is even. The probability that it is actually even is

\(\mathrm{S}=\{1,2,3,4,5,6,7\}\)
\(E_{1}=\) An even number is picked, \(E_{2}=\) An odd number is picked
\(P\left(E_{1}\right)=\frac{3}{7}, P\left(E_{2}\right)=\frac{4}{7}\)
\(E: A\) man reports an even number
\(P\left(E \mid E_{1}\right)=\frac{2}{3}\)
\(P\left(E \mid E_{2}\right)=\frac{1}{3}\)
Required probability \(=P\left(E_{1} \mid E\right)\)
\(=\frac{P\left(E \mid E_{1}\right) P\left(E_{1}\right)}{P\left(E \mid E_{1}\right) P\left(E_{1}\right)+P\left(E \mid E_{2}\right) P\left(E_{2}\right)}\)
\(=\frac{\left(\frac{2}{3}\right)\left(\frac{3}{7}\right)}{\left(\frac{2}{3}\right)\left(\frac{3}{7}\right)+\left(\frac{1}{3}\right)\left(\frac{4}{7}\right)}=\frac{6}{6+4}=\frac{3}{5}\)