Answer :
Let's analyze the given probabilities step-by-step.
### Total Participants
First, we need to know the total number of participants in the race. There are 12 women and 10 men.
Total number of participants = 12 (women) + 10 (men) = 22
### Probability Calculations
We are asked for the probability of specific combinations of the top three finishers:
#### (a) All Men
To find the probability that the top three finishers are all men:
1. The first finisher is a man: The probability is [tex]\( \frac{10}{22} \)[/tex].
2. The second finisher is also a man: Now there are 9 men left out of 21 participants. The probability is [tex]\( \frac{9}{21} \)[/tex].
3. The third finisher is also a man: Now there are 8 men left out of 20 participants. The probability is [tex]\( \frac{8}{20} \)[/tex].
Multiplying these probabilities together we get:
[tex]\[ \text{Probability (all men)} = \frac{10}{22} \times \frac{9}{21} \times \frac{8}{20} \approx 0.0779 \][/tex]
#### (b) All Women
To find the probability that the top three finishers are all women:
1. The first finisher is a woman: The probability is [tex]\( \frac{12}{22} \)[/tex].
2. The second finisher is also a woman: Now there are 11 women left out of 21 participants. The probability is [tex]\( \frac{11}{21} \)[/tex].
3. The third finisher is also a woman: Now there are 10 women left out of 20 participants. The probability is [tex]\( \frac{10}{20} \)[/tex].
Multiplying these probabilities together we get:
[tex]\[ \text{Probability (all women)} = \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \approx 0.1429 \][/tex]
#### (c) 2 Men and 1 Woman
To find the probability that the top three finishers are 2 men and 1 woman, there are several possible sequences: Man-Man-Woman, Man-Woman-Man, and Woman-Man-Man.
For each sequence, we compute the probabilities and sum them up:
1. Man-Man-Woman:
[tex]\[ \left( \frac{10}{22} \times \frac{9}{21} \times \frac{12}{20} \right) \][/tex]
2. Man-Woman-Man:
[tex]\[ \left( \frac{10}{22} \times \frac{12}{21} \times \frac{9}{20} \right) \][/tex]
3. Woman-Man-Man:
[tex]\[ \left( \frac{12}{22} \times \frac{10}{21} \times \frac{9}{20} \right) \][/tex]
Summing these probabilities:
[tex]\[ \text{Probability (2 men, 1 woman)} = \left( \frac{10}{22} \times \frac{9}{21} \times \frac{12}{20} \right) + \left( \frac{10}{22} \times \frac{12}{21} \times \frac{9}{20} \right) + \left( \frac{12}{22} \times \frac{10}{21} \times \frac{9}{20} \right) \approx 0.3506 \][/tex]
#### (d) 1 Man and 2 Women
Similarly, to find the probability that the top three finishers are 1 man and 2 women, there are several possible sequences: Man-Woman-Woman, Woman-Man-Woman, and Woman-Woman-Man.
For each sequence, we compute the probabilities and sum them up:
1. Man-Woman-Woman:
[tex]\[ \left( \frac{10}{22} \times \frac{12}{21} \times \frac{11}{20} \right) \][/tex]
2. Woman-Man-Woman:
[tex]\[ \left( \frac{12}{22} \times \frac{10}{21} \times \frac{11}{20} \right) \][/tex]
3. Woman-Woman-Man:
[tex]\[ \left( \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \right) \][/tex]
Summing these probabilities:
[tex]\[ \text{Probability (1 man, 2 women)} = \left( \frac{10}{22} \times \frac{12}{21} \times \frac{11}{20} \right) + \left( \frac{12}{22} \times \frac{10}{21} \times \frac{11}{20} \right) + \left( \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \right) \approx 0.4286 \][/tex]
### Summarized Results
(a) The probability that all top three finishers are men is approximately 0.0779.
(b) The probability that all top three finishers are women is approximately 0.1429.
(c) The probability that two of the top three finishers are men and one is a woman is approximately 0.3506.
(d) The probability that one of the top three finishers is a man and two are women is approximately 0.4286.
### Total Participants
First, we need to know the total number of participants in the race. There are 12 women and 10 men.
Total number of participants = 12 (women) + 10 (men) = 22
### Probability Calculations
We are asked for the probability of specific combinations of the top three finishers:
#### (a) All Men
To find the probability that the top three finishers are all men:
1. The first finisher is a man: The probability is [tex]\( \frac{10}{22} \)[/tex].
2. The second finisher is also a man: Now there are 9 men left out of 21 participants. The probability is [tex]\( \frac{9}{21} \)[/tex].
3. The third finisher is also a man: Now there are 8 men left out of 20 participants. The probability is [tex]\( \frac{8}{20} \)[/tex].
Multiplying these probabilities together we get:
[tex]\[ \text{Probability (all men)} = \frac{10}{22} \times \frac{9}{21} \times \frac{8}{20} \approx 0.0779 \][/tex]
#### (b) All Women
To find the probability that the top three finishers are all women:
1. The first finisher is a woman: The probability is [tex]\( \frac{12}{22} \)[/tex].
2. The second finisher is also a woman: Now there are 11 women left out of 21 participants. The probability is [tex]\( \frac{11}{21} \)[/tex].
3. The third finisher is also a woman: Now there are 10 women left out of 20 participants. The probability is [tex]\( \frac{10}{20} \)[/tex].
Multiplying these probabilities together we get:
[tex]\[ \text{Probability (all women)} = \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \approx 0.1429 \][/tex]
#### (c) 2 Men and 1 Woman
To find the probability that the top three finishers are 2 men and 1 woman, there are several possible sequences: Man-Man-Woman, Man-Woman-Man, and Woman-Man-Man.
For each sequence, we compute the probabilities and sum them up:
1. Man-Man-Woman:
[tex]\[ \left( \frac{10}{22} \times \frac{9}{21} \times \frac{12}{20} \right) \][/tex]
2. Man-Woman-Man:
[tex]\[ \left( \frac{10}{22} \times \frac{12}{21} \times \frac{9}{20} \right) \][/tex]
3. Woman-Man-Man:
[tex]\[ \left( \frac{12}{22} \times \frac{10}{21} \times \frac{9}{20} \right) \][/tex]
Summing these probabilities:
[tex]\[ \text{Probability (2 men, 1 woman)} = \left( \frac{10}{22} \times \frac{9}{21} \times \frac{12}{20} \right) + \left( \frac{10}{22} \times \frac{12}{21} \times \frac{9}{20} \right) + \left( \frac{12}{22} \times \frac{10}{21} \times \frac{9}{20} \right) \approx 0.3506 \][/tex]
#### (d) 1 Man and 2 Women
Similarly, to find the probability that the top three finishers are 1 man and 2 women, there are several possible sequences: Man-Woman-Woman, Woman-Man-Woman, and Woman-Woman-Man.
For each sequence, we compute the probabilities and sum them up:
1. Man-Woman-Woman:
[tex]\[ \left( \frac{10}{22} \times \frac{12}{21} \times \frac{11}{20} \right) \][/tex]
2. Woman-Man-Woman:
[tex]\[ \left( \frac{12}{22} \times \frac{10}{21} \times \frac{11}{20} \right) \][/tex]
3. Woman-Woman-Man:
[tex]\[ \left( \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \right) \][/tex]
Summing these probabilities:
[tex]\[ \text{Probability (1 man, 2 women)} = \left( \frac{10}{22} \times \frac{12}{21} \times \frac{11}{20} \right) + \left( \frac{12}{22} \times \frac{10}{21} \times \frac{11}{20} \right) + \left( \frac{12}{22} \times \frac{11}{21} \times \frac{10}{20} \right) \approx 0.4286 \][/tex]
### Summarized Results
(a) The probability that all top three finishers are men is approximately 0.0779.
(b) The probability that all top three finishers are women is approximately 0.1429.
(c) The probability that two of the top three finishers are men and one is a woman is approximately 0.3506.
(d) The probability that one of the top three finishers is a man and two are women is approximately 0.4286.
