In answering a question on a multiple choice test, a student either knows the answer or guesses. Let be the probability that he knows the answer and be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability . What is the probability that the student knows the answer given that he answered it correctly?

Solution:

Let represent the situation in which the student knows the solution, represent the situation in which the student guesses the answer, and represent the situation in which the answer is accurate.

Then

And

Also (correct answer given that he knows)

And (correct answer given that he guesses)

Now, the probability that he knows the answer is if the answer is accurate.

By using Bayes’ theorem, we have:

We get the following results by substituting the values: