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

The correct option is C

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

Explanation for the correct answer:

Step 1: According to the question.

Let $E$ be the event that the student answers correctly

Let $A$ be the event that student knows the answer

Let $B$ be the event that student does not know the correct answer.

Thus, $P\left(A\right)=p$

then, $P\left(B\right)=1-p$

Step 2: Finding the probability values.

The number of alternatives = $5$

Therefore

$$\begin{gathered} P\left(E|A\right)=\frac{5}{5} \\ =1 \end{gathered}$$

and $P\left(E|B\right)=\frac{1}{5}$

Step 3: Finding the conditional probability.

The probability that the students tick the answer correctly is equal to the probability that the students does not tick the answer randomly.

The conditional probability is as

$$\begin{gathered} \Rightarrow \frac{\left[P\left(A\right)P\left(E\right|A\right]}{P\left(A\right)P\left(E\right|A)+P(B\left)P\right(E\left|B\right)} \\ \Rightarrow \frac{\left(p\right)\left(1\right)}{\left(p\right)\left(1\right)+(1-p){\displaystyle \frac{1}{5}}} \\ \Rightarrow \frac{p}{p+{\displaystyle \frac{1-p}{5}}} \\ \Rightarrow \frac{p\left(5\right)}{5p+1-p} \\ \Rightarrow \frac{5p}{1+4p} \end{gathered}$$

Hence, option (C) $\frac{\mathbf{5}\mathbf{p}}{\mathbf{4}\mathbf{p}\mathbf{+}\mathbf{1}}$ is the correct answer.