Question

A student answers a multiple choice question with $5$ alternatives, of which exactly one is correct. The probability that he knows the correct answer is $p,0<p<1$. If he does not know the correct answer, he randomly ticks one answer. Given that he has answered the question correctly, the probability that he did not tick the answer randomly, is

• A. $\frac{3p}{4p+3}$
• B. $\frac{5p}{3p+2}$
• C. $\frac{5p}{4p+1}$
• D. $\frac{4p}{3p+1}$

Answer 1

Answer: (C)

$\frac{5p}{4p+1}$

Let the no. of alternatives be denoted as ${}^{\prime }{p}^{\prime }$

The question contains $5$ alternatives.

$\Rightarrow$ Let event $E_1=$ the answer that he ticked randomly.

$\Rightarrow$ Let event $E_2=$ the answer that he knows

$\Rightarrow A=$ answered correctly.

$\Rightarrow p\left(E_1\right)=\frac{5possibleanswers}{5}$

$=\frac{5}{5}$

$=1$

$\Rightarrow p\left(E_2\right)=\frac{1}{5}$

Probability that the ticked answer is wrong $=\frac{4}{5}$

$\therefore \frac{p\left(E_1\right)p\left(\frac{A}{E_1}\right)}{p\left(E_2\right)p\left({\frac{A}{E}}_2\right)+p\left(E_1\right)p\left({\frac{A}{E}}_1\right)}$

$=\frac{p\times 1}{\frac{4}{5}\times p+\frac{1}{5}\times 1}$

$=\frac{5p}{4p+1}$

Hence, the answer is $\frac{5p}{4p+1}.$