Question

In a multiple-choice test, an examinee either knows the correct answer with probability $p$, or guesses with probability $1-p$. The probability of answering a question correctly is $\frac{1}{m}$, if he or she merely guesses. If the examinee answers a question correctly, the probability that he or she really knows the answer is

• A. $\frac{mp}{1+mp}$
• B. $\frac{mp}{1+(m-1)p}$
• C. $\frac{(m-1)p}{1+(m-1)p}$
• D. $\frac{(m-1)p}{1+mp}$

Answer 1

Answer: (B)

$\frac{mp}{1+(m-1)p}$

Let $P\left(A\right)=$ Probability that he knows the answer and $P\left(B\right)=$ probability that answer is correct.

$P\left(B\right)=p.1+(1-p).\frac{1}{m}=\frac{1+(m-1)p}{m}$

$P\left(A|B\right)=\frac{P(A\cap B)}{P\left(B\right)}=\frac{p.1}{\frac{1+(m-1)p}{m}}=\frac{pm}{1+(m-1)p}$