If odds against solving a question by three students are 2: 1,5: … - Vidyadip

MCQ

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
$\frac{24}{56}$
✓
$\frac{25}{56}$
D
None of these

✓

Answer

Correct option: C.

$\frac{25}{56}$

(C)
The probability of solving the question by these three students are $\frac{1}{3}, \frac{2}{7}$ and $\frac{3}{8}$ respectively.
$\therefore P ( A )=\frac{1}{3} ; P ( B )=\frac{2}{7} ; P ( C )=\frac{3}{8}$
Then, probability of question solved by only one student $= P ( A \overline{ B } \overline{ C }$ or $\overline{ A } B \overline{ C }$ or $\overline{ A } \overline{ B } C )$
$= P ( A ) P (\overline{ B }) P (\overline{ C })+ P (\overline{ A }) P ( B ) P (\overline{ C }) + P (\overline{ A }) P (\overline{ B }) P ( C )$
$=\frac{1}{3} \cdot \frac{5}{7} \cdot \frac{5}{8}+\frac{2}{3} \cdot \frac{2}{7} \cdot \frac{5}{8}+\frac{2}{3} \cdot \frac{5}{7} \cdot \frac{3}{8}$
$=\frac{25+20+30}{168}=\frac{25}{56}$

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