Five different written driving tests are administered by the Motor Vehicle Department. One of these five tests is selected at random for each applicant for a driver's license. A group consisting of two women and three men apply for a license.
(a) What is the probability that exactly two of the five will take the same test?
(b) What is the probability that the two women will take the same test?
a) 5C2 * (1/5)^2 * (4/5)^3 = 0.2048
b)???
To solve these probability problems, we need to first calculate the total number of possible outcomes and then determine the favorable outcomes. Let's go step by step.
(a) To find the probability that exactly two of the five will take the same test, we need to consider all the possible scenarios where two people take the same test while the other three people take different tests.
Step 1: Calculate the total number of possible outcomes:
Since there are five different tests, each applicant can choose one of the five tests independently. So, each applicant has 5 options. Therefore, the total number of possible outcomes is 5 x 5 x 5 x 5 x 5 = 5^5 = 3125.
Step 2: Calculate the favorable outcomes:
The two people who take the same test can be any combination of the five applicants, and the test they take can be any of the five tests. We can consider the two women taking the same test as an example.
The favorable outcomes can be calculated as follows:
Number of ways to choose two women out of the total of five applicants: C(5, 2) = 5! / [(2!)(5-2)!] = 5! / (2!)(3!) = (5 x 4) / (2 x 1) = 10.
Number of ways to choose one test out of the five available tests: C(5, 1) = 5! / [(1!)(5-1)!] = 5! / (1!)(4!) = (5 x 4 x 3 x 2 x 1) / (4 x 3 x 2 x 1) = 5.
Number of ways to assign the two chosen women to the chosen test: 2! = 2.
Therefore, the favorable outcomes for exactly two of the five taking the same test is 10 x 5 x 2 = 100.
Step 3: Calculate the probability:
The probability is given by the ratio of favorable outcomes to the total number of outcomes.
Probability = Favorable outcomes / Total outcomes = 100 / 3125 = 0.032.
So, the probability that exactly two of the five will take the same test is 0.032 or 3.2%.
(b) To find the probability that the two women will take the same test, we need to consider the same logic as before, but now only focusing on the two women.
Step 1: Calculate the total number of possible outcomes (same as before): 5^5 = 3125.
Step 2: Calculate the favorable outcomes:
The two women can take the same test in only one way (since there are only two women). The test they take can be any of the five tests (five options).
Therefore, the favorable outcomes for the two women taking the same test is 1 x 5 = 5.
Step 3: Calculate the probability:
Probability = Favorable outcomes / Total outcomes = 5 / 3125 = 0.0016.
So, the probability that the two women will take the same test is 0.0016 or 0.16%.