Three critics review a book. For the three critics, the odds in favour of the book are \((5: 2),(4: 3)\) and \((3: 4)\) respectively. The probability that the majority is in favour of the book is

\(P\left(C_1\right)=\frac{5}{5+2}=\frac{5}{7}, P\left(C_2\right)=\frac{4}{4+3}=\frac{4}{7}, P\left(C_3\right)=\) \(\frac{3}{3+4}=\frac{3}{7}\)
now, required probability
\(=P\left(C_1\right) \cdot P\left(C_2\right) \cdot P\left(\bar{C}_3\right)+P\left(\bar{C}_1\right) \cdot P\left(C_2\right) P\left(C_3\right)\) \(+~P\left(C_1\right) \cdot P\left(\bar{C}_2\right) \cdot P\left(C_3\right)+P\left(C_1\right) \cdot P\left(C_2\right) \cdot P\left(C_3\right) \)
\( =\frac{5}{7} \cdot \frac{4}{7} \cdot \frac{4}{7}+\frac{2}{7} \cdot \frac{4}{7} \cdot \frac{3}{7}+\frac{5}{7} \cdot \frac{3}{7} \cdot \frac{3}{7}+\frac{5}{7} \cdot \frac{4}{7} \cdot \frac{3}{7} \)
\( =\frac{80+24+45+60}{343}=\frac{209}{343}\)