Question

An instructor gives his class a set of 18 problems with the information that the next quiz will consist of a random selection of 9 of them. If a student has figured out how to do 13 of the problems, what is the probability the he or she will answer correctly?

Answer 1

`\frac{{13 \choose 9}}{{18 \choose 9}}`

Explanation:

The student will answer correctly the quiz if he/she knows the answer to all of the quiz problems. The quiz consists of 9 problems out of the set of 18 problems. That means the professor could create
`{18 \choose 9}` different quizzes (which are the ways to choose a group of 9 problems out of the 18 problems). The student however doesn't know the answer to all of those different quizzes. He/she only knows the answer to the quiz if the 9 problems are all taken from the 13 problems he/she has already figured out. The number of quizzes that could be created, taking all 9 problems from the 13 problems he/she has already figure out is
`{13 \choose 9}`. So, the student is only able to answer
`{13 \choose 9}` of all possible quizzes the professor could come up with. Therefore the probability that the student answers correctly the quiz is
`\frac{{13 \choose 9}}{{18 \choose 9}}`