Seven different written driving tests are administered by the Motor Vehicle Department. One of these 7 tests is selected at random for each applicant for a driver's license.If a group consisting of two women and three men apply for a license, what is the probability that exactly two of the five will take the same test?

Question

Seven different written driving tests are administered by the Motor Vehicle Department. One of these 7 tests is selected at random for each applicant for a driver's license.If a group consisting of two women and three men apply for a license, what is the probability that  exactly two of the five will take the same test?

jmark · Answer

Count the number of ways you can assign test to the people and then divide by the total number of ways (which is 7^5). There are 4 way to choose which test is going to be done by two given applicants who take the same test. There are Binomial[5,2] ways to choose these two applicants. The remaining 3 applicants must be assigned any one of the 3 other tests. There are 3^3 ways for this. Now, 7*Binomial[5,2]*3^3 is not the correct answer, because of the double counting of the cases where 2 out of 3 of the remaining group of 3 are also assigned the same test. Depending on how the question is to be interpreted, such cases should not be counted at all, or they should be counted once. So, you need to compute the number of these cases and then subtract that once or twice from the above expression. You can also try a more formal approach using the principle of inclusion and exclusion. Edurite Google plus official page: https://plus.google.com/+EduriteScoreMore/