Question

If odds against solving a question by three students are 2 : 1, $5 : 2$ and $5 : 3$ respectively, then probability that the question is solved only by one student is

• A. $\frac { 31 } { 56 }$
• B. $\frac { 24 } { 56 }$
• C. $\frac { 25 } { 56 }$
• D. None of these

Answer 1

Answer: (C)

$\frac { 25 } { 56 }$

Answer 2

The probability of solving the question by these three students are $\frac { 1 } { 3 } , \frac { 2 } { 7 }$ and $\frac { 3 } { 8 }$ respectively.

$P ( A ) = \frac { 1 } { 3 }$ ; $P ( B ) = \frac { 2 } { 7 }$; $P ( C ) = \frac { 3 } { 8 }$

Then probability of question solved by only one student

$= P ( A \bar { B } \bar { C }$ or $\bar { A } B \bar { C }$ or $\bar { A } \bar { B } C )$

$= P ( A ) P ( \bar { B } ) P ( \bar { C } ) + P ( \bar { A } ) P ( B ) P ( \bar { C } ) + P ( \bar { A } ) P ( \bar { B } ) P ( C )$

$= \frac { 1 } { 3 } \cdot \frac { 5 } { 7 } \cdot \frac { 5 } { 8 } + \frac { 2 } { 3 } \cdot \frac { 2 } { 7 } \cdot \frac { 5 } { 8 } + \frac { 2 } { 3 } \cdot \frac { 5 } { 7 } \cdot \frac { 3 } { 8 }$ $= \frac { 25 + 20 + 30 } { 168 }$ $= \frac { 25 } { 56 }$.