Question
A race consists of 11 women and 7 men. Find the following probabilities for the top three finishers
Answers
GPT 3.5
Let's define the random variables X and Y such that X represents the number of women in the top 3 finishers, and Y represents the number of men in the top 3 finishers.
Since there are 11 women and 7 men, the total number of ways to choose the top 3 finishers is given by 18C3 = 816.
1. P(X = 3) - probability that all top 3 finishers are women
There are 11C3 ways to choose 3 women from the 11, and since there are no men in the top 3 finishers, the number of ways to choose 0 men from the 7 is 7C0. Therefore, P(X = 3) = (11C3 * 7C0) / 18C3 = (165 * 1) / 816 = 165 / 816 ≈ 0.2022
2. P(X = 2) - probability that 2 of the top 3 finishers are women
There are 11C2 ways to choose 2 women from the 11, and 7C1 ways to choose 1 man from the 7. Therefore, P(X = 2) = (11C2 * 7C1) / 18C3 = (55 * 7) / 816 = 385 / 816 ≈ 0.4723
3. P(X = 1) - probability that 1 of the top 3 finishers is a woman
There are 11 ways to choose 1 woman from the 11, and 7C2 ways to choose 2 men from the 7. Therefore, P(X = 1) = (11 * 21) / 816 = 231 / 816 ≈ 0.2831
4. P(Y = 3) - probability that all top 3 finishers are men
Similarly, P(Y = 3) = (11C0 * 7C3) / 18C3 = (1 * 35) / 816 = 35 / 816 ≈ 0.0429
5. P(Y = 2) - probability that 2 of the top 3 finishers are men
P(Y = 2) = (11C1 * 7C2) / 18C3 = (11 * 21) / 816 = 231 / 816 ≈ 0.2831
6. P(Y = 1) - probability that 1 of the top 3 finishers is a man
P(Y = 1) = (11C2 * 7C1) / 18C3 = (55 * 7) / 816 = 385 / 816 ≈ 0.4723
These are the probabilities for the top three finishers in the race.
Since there are 11 women and 7 men, the total number of ways to choose the top 3 finishers is given by 18C3 = 816.
1. P(X = 3) - probability that all top 3 finishers are women
There are 11C3 ways to choose 3 women from the 11, and since there are no men in the top 3 finishers, the number of ways to choose 0 men from the 7 is 7C0. Therefore, P(X = 3) = (11C3 * 7C0) / 18C3 = (165 * 1) / 816 = 165 / 816 ≈ 0.2022
2. P(X = 2) - probability that 2 of the top 3 finishers are women
There are 11C2 ways to choose 2 women from the 11, and 7C1 ways to choose 1 man from the 7. Therefore, P(X = 2) = (11C2 * 7C1) / 18C3 = (55 * 7) / 816 = 385 / 816 ≈ 0.4723
3. P(X = 1) - probability that 1 of the top 3 finishers is a woman
There are 11 ways to choose 1 woman from the 11, and 7C2 ways to choose 2 men from the 7. Therefore, P(X = 1) = (11 * 21) / 816 = 231 / 816 ≈ 0.2831
4. P(Y = 3) - probability that all top 3 finishers are men
Similarly, P(Y = 3) = (11C0 * 7C3) / 18C3 = (1 * 35) / 816 = 35 / 816 ≈ 0.0429
5. P(Y = 2) - probability that 2 of the top 3 finishers are men
P(Y = 2) = (11C1 * 7C2) / 18C3 = (11 * 21) / 816 = 231 / 816 ≈ 0.2831
6. P(Y = 1) - probability that 1 of the top 3 finishers is a man
P(Y = 1) = (11C2 * 7C1) / 18C3 = (55 * 7) / 816 = 385 / 816 ≈ 0.4723
These are the probabilities for the top three finishers in the race.