# On a question chosen at random, what is the probability that the student answers it correctly?

A student takes a multiple choice exam where each question has five possible answers. He answers correctly if he knows the answer, otherwise he guesses at random. Suppose he knows the answer to $70$% of the questions.

Question 1

On a question chosen at random, what is the probability the student answers it correctly?

We know $P(\text{know the answer}) = 0.7$. On a question chosen at random, he either knows the answer or he doesn't. If he does, then $P(\text{answer correctly}|\text{know correct answer})= 1$. If he doesn't, then $P(\text{answer correctly}|\text{doesn't know correct answer})= 0.7(0.3)^4$. I don't know how to continue the answer.

Question 2

Given that he did answer correctly, what is the probability that he actually knew the correct answer?

All I can think of is the following conditional probability.

\begin{align} P(\text{know correct answer}|\text{answer correctly}) & = P(\text{know correct answer}|\text{answer correctly}) \\ & =\frac{P(\text{know correct answer and answer correctly})}{P(\text{answer correctly)}}. \\ \end{align}

• If the student doesn't know the answer the chance of guessing the correct one is $\frac{1}{5}$ rather than $0.7\cdot(0.3)^4$. – Julian May 23 '13 at 19:18

For the first one, drawing a probability tree might help.

At first, you have the two branches of whether he knows the answer (0.7) and he doesn't know the answer (0.3).

If he knows the answer, then he answers correctly (1), so that, the probability that he knows and answer AND answers correctly is (0.7*1 =) 0.7.

If he doesn't know the answer, he has a probability of (1/5 =) 0.2 of getting the right answer, so that the probability that he doesn't know the answer AND answers correctly becomes: (0.3*0.2 =) 0.06

The rest should be a little easier.

Consider a typical run of 100 questions.

How many will he know the answer for? Obviously, he gets all of these right.

How many are left to guess about? How many of these will he get right?

How many correct all together from the entire 100 question run?

Your logic for the second question is correct, given the correct numbers to insert...