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

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Aicharya · Accepted Answer

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

Three critics review a book. For the three critics, the odds in favour of the book are $(5: 2),(4: 3)$ and…

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