The odds that a book will be reviewed favorably in three independent critics are 5 to 2, 4 to 3 and 3 to 4 respectively. The probability that of the three reviews majority will be favorable  is $\mathrm{\lambda }$ then 343$\mathrm{\lambda }$ =

Let ${\mathrm{p}}_1,{\mathrm{p}}_2,{\mathrm{p}}_3$ denote the probabilities that the first , second and third critics reviews the book favorably now ${\mathrm{p}}_1=\frac{5}{7},{\mathrm{p}}_2=\frac{4}{7},{\mathrm{p}}_3=\frac{3}{7},\stackrel{\mathrm{\_}}{{\mathrm{p}}_1}=\frac{2}{7},\stackrel{\mathrm{\_}}{{\mathrm{p}}_2}=\frac{3}{7},\stackrel{\mathrm{\_}}{{\mathrm{p}}_3}=\frac{4}{7}$ The probability of the majority will be favorable $={\mathrm{p}}_1{\mathrm{p}}_2\stackrel{\mathrm{\_}}{{\mathrm{p}}_3}+{\mathrm{p}}_1\stackrel{\mathrm{\_}}{{\mathrm{p}}_2}{\mathrm{p}}_3+\stackrel{\mathrm{\_}}{{\mathrm{p}}_1}{\mathrm{p}}_2{\mathrm{p}}_3+{\mathrm{p}}_1{\mathrm{p}}_2{\mathrm{p}}_3$
$\begin{array}{l}=\frac{5}{7}\cdot \frac{4}{7}\cdot \frac{4}{7}+\frac{5}{7}\cdot \frac{3}{7}\cdot \frac{3}{7}+\frac{2}{7}\cdot \frac{4}{7}\cdot \frac{3}{7}+\frac{5}{7}\cdot \frac{4}{7}\cdot \frac{3}{7}=\frac{209}{343}=\mathrm{\lambda }\\ \therefore 343\mathrm{\lambda }=209\end{array}$