Final answer:
The probability that two men's names are chosen without replacement from a group of 7 women and 10 men is approximately 33.09%.
Step-by-step explanation:
The question asks us to find the probability that two men's names are selected without replacement from a group of 7 women and 10 men when two doses of a flu protection vaccine are available. To solve this problem, we use combinations to determine the probability.
First, we find the total number of ways to choose 2 people from the group of 17 (7 women + 10 men), which is a combination of 17 taken 2 at a time, denoted as C(17, 2). Next, we find the number of ways to choose 2 men from the 10 available, which is C(10, 2). The probability of selecting 2 men is then calculated by dividing the number of ways to choose 2 men by the total number of ways to choose 2 people.
The formula for combinations is C(n, k) = n! / (k!(n - k)!), where n is the total number of items, k is the number of items to choose, and ! denotes factorial. We use this to calculate:
- C(17, 2) = 17! / (2!(17 - 2)!) = 136
- C(10, 2) = 10! / (2!(10 - 2)!) = 45
The probability that the two names selected are men's names is C(10, 2) / C(17, 2) = 45 / 136 = 0.33088, or about 33.09%.