Question

The odds that a book will be reviewed favourably by three independent critics are 5 to 2, 4 to 3 and 3 to 4 respectively. The probability that of the three reviewers a majority will be favourable

• A. $\frac{134}{343}$
• B. $\frac{109}{343}$
• C. $\frac{209}{343}$
• D. $\frac{309}{343}$

Answer 1

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

Let the three critics be $A,B$ and $C$.
The odds that a book will be reviewed favorably by $A,B$ and $C$ are $5$ to $2,4$ to $3$ and $3$ to $4$ respectively.
i.e. $P\left(A\right)=\frac{5}{7},P\left(B\right)=\frac{4}{7},P\left(C\right)=\frac{3}{7}$
$\Rightarrow P\left(\overline{A}\right)=\frac{2}{7},P\left(\overline{B}\right)=\frac{3}{7},P\left(\overline{C}\right)=\frac{4}{7}$
To find the probability that a majority will be favorable is the same as the probability that at least two of them are favorable.
$\therefore$ the required probability
$=P(A\cap B\cap \overline{C})+P(A\cap \overline{B}\cap C)+P(\overline{A}\cap B\cap C)+P(A\cap B\cap C)=P\left(A\right)P\left(B\right)P\left(\overline{C}\right)+P\left(A\right)P\left(\overline{B}\right)P\left(C\right)+P\left(\overline{A}\right)P\left(B\right)P\left(C\right)+P\left(A\right)P\left(B\right)P\left(C\right)$
$=\frac{2}{7}\times \frac{4}{7}\times \frac{4}{7}+\frac{5}{7}\times \frac{3}{7}\times \frac{3}{7}+\frac{2}{7}\times \frac{4}{7}\times \frac{3}{7}+\frac{5}{7}\times \frac{4}{7}\times \frac{3}{7}=\frac{209}{343}$