Question

Three critics review a book. Odds in favour of the book are

5 : 2, 4 : 3 and 3 : 4 respectively for the three critics. The

probability that majority are in favour of the book is

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

Answer 1

Answer: (D)

$\frac { 209 } { 343 }$

Answer 2

Probability that the first critic favours the book, P(E1)

= $\frac { 5 } { 5 + 2 } = \frac { 5 } { 7 }$

Probability that the second critic favours the book, P(E2)

= $\frac { 4 } { 4 + 3 } = \frac { 4 } { 7 }$

Probability that the third critic favours the book, P(E3)

= $\frac { 3 } { 3 + 4 } = \frac { 3 } { 7 }$

Majority will be in favour if at least two critics favour the book

= P(E1 Ē E2 Ē $\overline { \mathrm { E } } _ { 3 }$ ) + P(E1 Ē $\overline { \mathrm { E } } _ { 2 }$ Ē E3) + P

( $\overline { \mathrm { E } } _ { 1 }$ Ē E2 Ē E3) + P(E1 Ē E2 Ē E3)

= P(E1) P(E2) P( $\overline { \mathrm { E } } _ { 3 }$ ) + P(E1) P( $\overline { \mathrm { E } } _ { 2 }$ ) P(E3) +

P( $\overline { \mathrm { E } } _ { 1 }$ Ē E2 Ē E3) + P(E1) P(E2) P(E3)

= $\frac { 5 } { 7 } \times \frac { 4 } { 7 } \times \left( 1 - \frac { 3 } { 7 } \right) + \frac { 5 } { 7 } \times \left( 1 - \frac { 4 } { 7 } \right) \times \frac { 3 } { 7 } + \left( 1 - \frac { 5 } { 7 } \right) \times$ $\frac { 4 } { 7 } \times \frac { 3 } { 7 } + \frac { 5 } { 7 } \times \frac { 4 } { 7 } \times \frac { 3 } { 7 } = \frac { 209 } { 343 }$