# Probability of getting 2 questions correct on a test by students of different skill levels

Out of the 14 students taking a test, 5 are well prepared, 6 are adequately prepared and 3 are poorly prepared. There are 10 questions on the test paper. A well prepared student can answer 9 questions correctly, an adequately prepared student can answer 6 questions correctly and a poorly prepared student can answer only 3 questions correctly.

(a) If a randomly chosen student is asked two distinct randomly chosen questions from the test, what is the probability that the student will answer both questions correctly? Note: The student and the questions are chosen independently of each other. “Random” means that each individual student/each pair of questions is equally likely to be chosen. (b) Now suppose that a student is chosen at random and asked two randomly chosen questions from the exam, and moreover ݀݅݀ answer both questions correctly. Find the probability that the chosen student was well prepared.

I started by making 3 cases

Case 1: The student is well prepared.

The probability of picking a well prepared student is $\frac{5}{14}$, the probability that he got the first question right was $\frac{9}{10}$ and the probability of getting the second question correct will be $\frac{8}{9}$ Multiplying them we get $\frac{2}{7}$

I similarly proceeded with the other cases and added them up to get the answer as $\frac{31}{70}$ for the first part and $\frac{20}{31}$ for the second one.

However none of my answers match the correct ones... Please tell me where I went wrong and what is the correct answer, I was thinking in terms of the distinguishability of the 2 questions in the first one for which we might have to divide further by $2!$

How to do this question correctly?

• I think the question is ambiguous. Does it mean, as you seem to imagine, that (for the prepared student, say) there are $9$ questions he can answer and $1$ that he can not or does it mean that he has a $.9$ probability of getting any given question right? These are similar but not the same. – lulu Sep 1 '18 at 11:14
• What exactly is meant with "can answer"? – drhab Sep 1 '18 at 11:19
• I believe it is the former as it says that he can answer 9 question correctly. – K. Chopra Sep 1 '18 at 11:20
• Possibly the ability to answer only a certain number of questions. At least that's what I assumed. – K. Chopra Sep 1 '18 at 11:20
• Well, you also say that your answer is incorrect. What is the "official" answer? Is it consistent with the second interpretation I give? – lulu Sep 1 '18 at 11:21