Question

The odds that a book will be favorably 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 favorable?

• A. $\frac{209}{343}$
• B. $\frac{200}{343}$
• C. $\frac{109}{343}$
• D. none of these

Answer 1

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

Given,the odds that a book will be favorably reviewed by three independent critics are 5 to 2, 4 to 3 and 3 to 4
respectively.
$\therefore$The Probability that a book will be favorably reviewed by first independent critics, $P\left(A\right)=\frac{5}{7}$,
$\therefore$The Probability that a book will be favorably reviewed by second independent critic, $P\left(B\right)=\frac{4}{7}$,
$\therefore$The Probability that a book will be favorably reviewed by third independent critic, $P\left(C\right)=\frac{3}{7}$.
The probability that of the three reviews a majority will be favorable=

$P\left(A\right)P\left(B\right)P\left(C\right)+P\left(A\right)P\left(B\right)P\left(\overline{C}\right)+P\left(B\right)P\left(C\right)P\left(\overline{A}\right)+P\left(C\right)P\left(A\right)P\left(\overline{B}\right)$

$=\frac{5}{7}\times \frac{4}{7}\times \frac{3}{7}+(\frac{5}{7}\times \frac{4}{7}\times (1-\frac{3}{7}))+(\frac{4}{7}\times \frac{3}{7}\times (1-\frac{5}{7}))+(\frac{3}{7}\times \frac{5}{7}\times (1-\frac{4}{7}))$

$=\frac{(60+80+24+45)}{343}=\frac{209}{343}$