four different written driving tests are administered by the MVD. on of these four tests is selected at random for each applicant for a drivers test. if a group consisting of two women and three men apply for a license, what is the probability that

a. exactly two of the five will take the same test?

b. the two women will take the same test?

thank you for your help, i need to know how to set up this problem, it is very confusing

correction, on actually is one

To solve this problem, we need to understand the concept of probability and use it to determine the likelihood of certain outcomes.

First, let's determine the total number of possible outcomes. Since there are four different tests and each applicant may be assigned any of these tests, the total number of possible outcomes is 4 multiplied by 4 multiplied by 4 multiplied by 4 multiplied by 4, which is 4^5 = 1024.

Now, we can proceed to solve the given questions:

a. Exactly two of the five will take the same test:
To calculate the probability that exactly two of the five applicants will take the same test, we need to consider the different ways this can occur. There are two scenarios where this is possible:

Scenario 1:
- Two women take the same test.
- Three men take different tests.

Scenario 2:
- Two men take the same test.
- Two women take different tests.

For Scenario 1:
- To select two women to take the same test, we choose 2 out of 2 women, which can be done in C(2, 2) = 1 way.
- To assign a test to these two women, we have 4 choices.
- To assign tests to the three remaining men, each man has 4 choices.
Therefore, the total number of outcomes for Scenario 1 is 1 * 4 * 4 * 4 * 4 = 256.

For Scenario 2:
- To select two men to take the same test, we choose 2 out of 3 men, which can be done in C(3, 2) = 3 ways.
- To assign a test to these two men, we have 4 choices.
- To assign tests to the two women, each woman has 4 choices.
- The remaining man will have 3 choices.
Therefore, the total number of outcomes for Scenario 2 is 3 * 4 * 4 * 4 * 3 = 576.

Therefore, the total number of outcomes where exactly two of the five applicants take the same test is 256 + 576 = 832.

To calculate the probability, we divide the favorable outcomes (832) by the total number of outcomes (1024):
Probability = 832 / 1024 ≈ 0.8125 or 81.25%.

b. The two women will take the same test:
To calculate the probability that the two women will take the same test, we only need to consider Scenario 1 from the previous question. As we determined earlier, the number of outcomes for Scenario 1 is 256.

To calculate the probability, we divide the favorable outcomes (256) by the total number of outcomes (1024):
Probability = 256 / 1024 = 0.25 or 25%.