What is the probability that you choose the December 1st issue of Newsweek?

Question

Assume that there are 8 different issues of Sports Illustrated, 7 different issues of Time, and 4 different issues of Newsweek, including the December 1st issue, on the rack. You choose 4 of them at random. 

(1) What is the probability that exactly 2 are issues issue of Sports Illustrated?

(2) What is the probability that you choose the December 1st issue of Newsweek?

Answer 1

(1) The probability is the number of successful outcomes (in this case, exactly 2 Sports Illustrated issues are picked) divided the total number of outcomes. Since there are a total of 8 + 7 + 4 = 19, and you choose 4 of them, the total number of outcomes is the number of ways to choose 4 items from the 19 items, and this is Choose(19,4) or simply C(19, 4). It is not permutations, since the order you in which you select the magazines is irrelevant.

Now we need to know how many ways are there to select exactly 2 Sports illustrated issues: From the 8 Sports Illustrated magazines, we need to choose exactly two, so this is C(8, 2). And (which when doing things in a sequence is multiplication), we need two magazines that are not Sports Illustrated.  There are 11 magazines that are not Sports Illustrated, so there are C(11, 2) ways to pick those. So, the final answer is:

C(8, 2) * C(11, 2) / C(19, 4).

(2) To get the number of ways to choose the December 1st  issue of Newsweek when selecting 4, we need to pick that December 1st issue (only 1 way to do that) and now select any 3 other magazines of the 18 remaining magazines, which is C(18, 3). The total number of outcomes are the same, so the answer here is:

C(18, 3) / C(19, 4).