Question

The odds that a book will be favourably reviewed by three independent critics are $5$ to $2$, $4$ to $3$, and $3$ to $4$ respectively; what is the probability that of the three reviews a majority will be favourable?

Answer 1

$P_1,P_2\&P_3$ be probability of favorable review
$P_1=\frac{5}{7}{P}_3=\frac{3}{7}{P}_1^{{}^{\prime }}=\frac{2}{7}{P}_3^{{}^{\prime }}=\frac{4}{7}$
$P_2=\frac{4}{7}{P}_2^{{}^{\prime }}=\frac{3}{7}$
For majority at least two must be favorable
Required probability$=\frac{5}{7}\times \frac{3}{7}\times \frac{4}{7}+\frac{2}{7}\times \frac{4}{7}\times \frac{3}{7}+\frac{5}{7}\times \frac{3}{7}\times \frac{4}{7}+\frac{5}{7}\times \frac{4}{7}\times \frac{3}{7}$
$=\frac{60}{7^3}+\frac{24}{7^3}+\frac{60}{7^3}+\frac{60}{7^3}$
$=\frac{197}{343}$