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Sagot :
To find the probability that a randomly selected four-person committee consists of all boys from a group of 8 boys and 6 girls, we can follow these steps:
1. Determine the Total Number of Boys and Girls:
- There are 8 boys.
- There are 6 girls.
- Total students = 8 boys + 6 girls = 14 students.
2. Calculate the Total Number of Ways to Form a Four-Person Committee from 14 Students:
- The number of ways to choose 4 students out of 14 (regardless of gender) is given by the combination formula [tex]\(\binom{n}{k}\)[/tex], which is [tex]\(\frac{n!}{k!(n-k)!}\)[/tex], where [tex]\(n\)[/tex] is the total number of students and [tex]\(k\)[/tex] is the number of students to choose.
- Thus, the total number of ways to form a committee of 4 from 14 students is [tex]\(\binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14!}{4! \cdot 10!} = 1001\)[/tex].
3. Calculate the Number of Ways to Form a Four-Person Committee Comprised Entirely of Boys:
- The number of ways to choose 4 boys out of 8 (since we want the committee to consist only of boys) is also given by the combination formula.
- Thus, the number of ways to choose 4 boys out of 8 is [tex]\(\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} = 70\)[/tex].
4. Calculate the Probability:
- The probability of forming a four-person committee that consists entirely of boys is the ratio of the number of favorable outcomes (all boys) to the total number of possible outcomes (any gender).
- So, the probability is [tex]\(\frac{\text{Number of ways to choose 4 boys out of 8}}{\text{Total number of ways to choose 4 students out of 14}}\)[/tex] which simplifies to [tex]\[ \frac{70}{1001} \approx 0.06993006993006994. \][/tex]
5. Match the Probability with the Given Options:
- Comparing [tex]\(\frac{70}{1001}\)[/tex] to the given options:
- [tex]\(\frac{4}{1001}\)[/tex]
- [tex]\(\frac{15}{\sqrt{10001}}\)[/tex]
- [tex]\(\frac{10}{143}\)[/tex]
- [tex]\(\frac{133}{143}\)[/tex]
- Clearly, [tex]\(\frac{70}{1001}\)[/tex] matches [tex]\(\frac{10}{143}\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{\frac{10}{143}}\)[/tex].
1. Determine the Total Number of Boys and Girls:
- There are 8 boys.
- There are 6 girls.
- Total students = 8 boys + 6 girls = 14 students.
2. Calculate the Total Number of Ways to Form a Four-Person Committee from 14 Students:
- The number of ways to choose 4 students out of 14 (regardless of gender) is given by the combination formula [tex]\(\binom{n}{k}\)[/tex], which is [tex]\(\frac{n!}{k!(n-k)!}\)[/tex], where [tex]\(n\)[/tex] is the total number of students and [tex]\(k\)[/tex] is the number of students to choose.
- Thus, the total number of ways to form a committee of 4 from 14 students is [tex]\(\binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14!}{4! \cdot 10!} = 1001\)[/tex].
3. Calculate the Number of Ways to Form a Four-Person Committee Comprised Entirely of Boys:
- The number of ways to choose 4 boys out of 8 (since we want the committee to consist only of boys) is also given by the combination formula.
- Thus, the number of ways to choose 4 boys out of 8 is [tex]\(\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} = 70\)[/tex].
4. Calculate the Probability:
- The probability of forming a four-person committee that consists entirely of boys is the ratio of the number of favorable outcomes (all boys) to the total number of possible outcomes (any gender).
- So, the probability is [tex]\(\frac{\text{Number of ways to choose 4 boys out of 8}}{\text{Total number of ways to choose 4 students out of 14}}\)[/tex] which simplifies to [tex]\[ \frac{70}{1001} \approx 0.06993006993006994. \][/tex]
5. Match the Probability with the Given Options:
- Comparing [tex]\(\frac{70}{1001}\)[/tex] to the given options:
- [tex]\(\frac{4}{1001}\)[/tex]
- [tex]\(\frac{15}{\sqrt{10001}}\)[/tex]
- [tex]\(\frac{10}{143}\)[/tex]
- [tex]\(\frac{133}{143}\)[/tex]
- Clearly, [tex]\(\frac{70}{1001}\)[/tex] matches [tex]\(\frac{10}{143}\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{\frac{10}{143}}\)[/tex].
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