Question

Suppose that in one of your classes you are given 20 questions, and are told that the final exam will consist of 8 of them. Suppose that the 8 that are chosen for the final are selected randomly.

a) If you work out how to do 12 of the problems, what is the probability that you will be able to work at least 7 of those on the final?

b) How many problems do you need to have worked out to have a 90% probability of being able to solve all 8 of those on the final?

Answer 1

(a)

Let X is a random variable shows the number of problems that you will be able to work on the final. Here X has hypergeometric distribution with following parameters,

Population size, N = 20

Number of problems you work out, k = 12

Sample size, n=8

The probability that you will be able to work at least 7 of those on the final is

(b)

Let k shows the number of problems you work out. So,

It is given that

Here we need to find k such that following equation is true,

Since examiner selected 8 questions so k cannot be less than 8. Following table shows the value of right hand side of above equation for k=8,9,10....

k	(k! * 12!) / ((k-8)! *20!)
8	7.9384E-06
9	7.14456E-05
10	0.000357228
11	0.001309836
12	0.003929507
13	0.010216718
14	0.023839009
15	0.051083591
16	0.102167183
17	0.192982456
18	0.347368421
19	0.6
20	1

So you need to have worked out all 20 problems to have a 90% probability (actually 100%) of being able to solve all 8 of those on the final. For less than 20 problems you will have only 60% or less chance of being able to solve all 8 of those on the final.