# “Practice Exam” probabilities

A practice exam with 6 questions is prepared. 3 of which will be very similar to ones on the actual exam.

a) How many possible ways can the exam be written? (i.e. how big is the sample space of possible exams)?

b) You chose 4 questions to study randomly. What's the probability that you will have studied all three of the questions that will be on the actual exam?

c) What's the probability that you will have missed at least one of the problems on the actual exam?

d) What's the probability that you will have covered none of the problems on the actual exam?

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Are the types of questions just those similar to the actual exam and not? Is this the only kind of answers there. – JB King Feb 5 '13 at 22:09
That's all that's given... – user60852 Feb 5 '13 at 23:53

## 1 Answer

Given that there are two kinds of questions, those similar to the actual exam and those that aren't similar to the actual exam, here is how I'd approach each of these questions:

a) The exam can be written in $\binom{6}{3}$ ways which is 20 possibilities.

b) $\binom{4}{3}$ is 4 which would suggest there are 4 combinations that would be out of the 20 which is 1/5 or .2 or 20%.

c) This is the remainder from the previous case which is 4/5 or .8 or 80% as you'll either study all the problems or miss at least one of them.

d) Zero. By the pigeon-hole principle, there are only 2 questions left unstudied and there are 3 on the exam, thus at least one has to be on the exam.

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