Question

In a test an examine either gusses or copies or knows the answer to a multiple choice question with $m$ choices out of which exactly one is correct. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. The probability that his answer is correct given that the copied it, is 1/8. If the probability that he knew the answer to the question given that he correctly answered it is 120/141, find $m$

Answer 1

Let $G$ denote the event the examine guesses, $C$ the event that the examinee copies and $K$ the event that the examinee knows. Let $R$ denote the event that the answer is right. We have
$P\left(G\right)=\frac{1}{3},P\left(C\right)=\frac{1}{6}$
and $P\left(K\right)=1-\frac{1}{3}-\frac{1}{6}=\frac{1}{2}$
Also,
$P\left(R|G\right)=\frac{1}{m},P\left(R|C\right)=\frac{1}{8},P\left(R|K\right)=1$.
By Bayes' rule
$P\left(K|R\right)=\frac{24m}{16+25m}=\frac{120}{141}\Rightarrow m=5$