I'm currently trying to solve this probability question and I'm quite unsure if my answer is correct or not.
On a multiple-choice exam, there are $100$ questions each with $4$ possible choices.
A student is certain of the correct answer to each question with probability $0.6$ and guesses randomly among the $4$ choices otherwise.
i. What is the probability that the student correctly answers question #$1$?
ii. What is the probability that the student was certain of the answer to question #$1$ given that they got it correct?
My Solutions:
i. $(100\cdot4)\cdot(0.6/100)=400\cdot0.006=2.4$
ii. $2.4/100=0.024$
I think I might be a bit off.
Can anyone tell me if my calculations are correct?
Best Answer
Let's assume that "is certain of the correct answer" = "knows the correct answer".
Without assuming that, this "certainty" would be equivalent to "guessing randomly".
Split it into disjoint events, and add up their probabilities:
$$0.6+(1-0.6)\cdot\frac14=0.7$$
Let $A$ denote the event of the student being certain of the answer to question #$1$.
Let $B$ denote the event of the student correctly answering question #$1$.
Then:
$$P(A|B)=\frac{P(A \cap B)}{P(B)}=\frac{0.6}{0.7}=\frac67$$