Question

Three critics review a book. Odds in favor of the book are $5:2,4:3$ and $3:4$, respectively, for the three critics. The probability that majority are in favor of the book is

• A. $\frac{35}{49}$
• B. $\frac{125}{343}$
• C. $\frac{164}{343}$
• D. $\frac{209}{343}$

Answer 1

The correct option is D $\frac{209}{343}$

$P\left(A\right)=\frac{5}{7},P\left(B\right)=\frac{4}{7},P\left(C\right)=\frac{3}{7}$

Probability that majority are in favor of the book is

$=P\left(A\right)\times P\left(B\right)\times P\left(C\right)+P(A{)}^c\times P(B)\times P(C)+P(A)\times P(B{)}^c\times P\left(C\right)+P\left(A\right)\times P\left(B\right)\times P(C{)}^c$

$=(\frac{5}{7}\times \frac{4}{7}\times \frac{3}{7})+(\frac{2}{7}\times \frac{4}{7}\times \frac{3}{7})+(\frac{5}{7}\times \frac{3}{7}\times \frac{3}{7})+(\frac{5}{7}\times \frac{4}{7}\times \frac{4}{7})=\frac{209}{343}$