our different written driving tests are administered by the MVD. one 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?
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To solve these probability problems, we need to analyze the number of possible outcomes and the desired outcomes. Let's break down each question:
a. To find the probability that exactly two of the five will take the same test, we need to consider the different ways this can happen.
First, let's determine the total number of possible outcomes. Each applicant has four test options, so there are 4^5 = 1024 total possible outcomes. This is because for each applicant (five in total), there are four possible tests they can be assigned.
Now, let's determine the number of desired outcomes. To have exactly two of the five take the same test, we can approach this by considering two cases:
Case 1: Two women take the same test, and the three men take different tests.
The number of ways to choose two women out of the total two is (C(2,2) = 1). For each combination of two women, they can take the same test in four different ways (one of the four tests). The men can take a different test each, which can be done in (4 * 3 * 2) = 24 ways. Therefore, the number of desired outcomes for this case is 1 * 4 * 24 = 96.
Case 2: Two men take the same test, and the two women take different tests.
Similarly, the number of ways to choose two men out of the total three is (C(3,2) = 3). For each combination of two men, they can take the same test in four different ways. The women can take a different test each, which can be done in (4 * 3) = 12 ways. Therefore, the number of desired outcomes for this case is 3 * 4 * 12 = 144.
Now that we have the number of desired outcomes for each case, we can find the total number of desired outcomes by summing the two cases: 96 + 144 = 240.
Finally, the probability of exactly two of the five taking the same test is given by the ratio of desired outcomes to total outcomes: 240/1024 ≈ 0.2344 or approximately 23.44%.
b. To find the probability that the two women will take the same test, we only need to consider one case: the two women taking the same test, and the three men taking different tests. Based on the analysis above, we discovered that there are 96 desired outcomes for this case.
The total number of possible outcomes is still 1024.
Therefore, the probability that the two women will take the same test is 96/1024 = 3/32 ≈ 0.09375 or approximately 9.375%.