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 Question from Carla, a student: You are taking a multiple choice quiz  that consist in 3 questions, each question has 3 possible answers only one is correct. To complete the quiz you randomly guess the answer to each question. Find the probability of guessing  exactly 2 answer correctly. b) at least to answer correctly . c)less than two answer correctly

Hi Carla.

If there were just one question, then the probability of guessing correctly would be 1/3.

Since all the answers are independent (the answer to one question has no bearing on the answers to the others), then this is the case with each question, so the chances of guessing all answers correctly is 1/3 × 1/3 × 1/3 = 1/27. Independent choices are linked by multiplication.

To have exactly 2 answers correct, we have to think of which one is wrong: there are 3 questions and any single one could be wrong. The probability that the first question is wrong is 2/3. And we know that the probabilities of the other two being right is 1/3 each, so the probability of just the first question being wrong and the others right is 2/3 × 1/3 × 1/3 = 2/27. But this is just one of the three cases: 1/3 × 2/3 × 1/3 and 1/3 × 1/3 × 2/3 also both equal 2/27 each. So here we add the cases together: 2/27 + 2/27 + 2/27 = 6/27 = 2/9. So the answer to part (a) of your question is 2/9.

To solve (b) Consider that (b) is the same as (a) with "all answers right" added in. So you can simply add answer (a) to the chance of guessing all answers correctly.

To solve (c) it might help you to think of the equivalent problem: what is the probability of getting 0 correct plus the probability of getting 1 correct?

Hope this helps,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.