Probability Question

Question

On a multiple-choice examination with four choices for each question, a student either knows the answer to a question or marks it randomly. The probability that they know the answer for any problem is

If a question was marked correctly, what is the probability they knew the answer?

Answer 1

You didn't give the probability that they know the answer to a problem, but we'll call that P.

If a question is marked correctly, then either they knew the answer with probability P or they didn't know, and guessed. The probability that they have to guess is 1-P and the probability that they get it right when guessing is 1/4. So that probability that they got a correct answer is:

P + (1-P)(1/4)

We know the probability that they get an answer correct, given that they know the answer (it's probability 1!), but we're looking for the conditional probability that they know the answer, given that it was marked correct.

Let's call the event that they get a correct answer C. Let's call the event of knowing the answer K.

So we know that p(C) = P + (1-P(1/4) and that p(K) = P and p(C | K) = 1 (as above), but we're looking for p(K | C). Given what we know, it can be found by calculating:

p(P | [P + {1-P}{1/4}])

So you just need to put in the value of P and you're good to go!

Hope that helps!