To determine the probability that a four-person committee chosen from a group of eight boys and six girls consists entirely of boys, we'll follow these steps:
1. Calculate the total number of boys and girls:
- There are 8 boys and 6 girls.
- Therefore, the total number of students is:
[tex]\[
8 + 6 = 14
\][/tex]
2. Determine the number of ways to choose 4 boys out of the 8 boys:
- The number of ways to choose 4 boys from 8 is calculated using combinations:
[tex]\[
\binom{8}{4}
\][/tex]
- This gives us 70 ways:
[tex]\[
\binom{8}{4} = 70
\][/tex]
3. Determine the total number of ways to form a committee of 4 students from all 14 students:
- The number of ways to choose 4 students from 14 is calculated using combinations:
[tex]\[
\binom{14}{4}
\][/tex]
- This gives us 1001 ways:
[tex]\[
\binom{14}{4} = 1001
\][/tex]
4. Calculate the probability that the committee consists of all boys:
- The probability is the ratio of the number of all-boy committees to the total number of committees:
[tex]\[
\text{Probability} = \frac{\text{Number of ways to choose 4 boys from 8 boys}}{\text{Total number of ways to choose 4 students from 14 students}}
\][/tex]
- Substituting the numbers we have:
[tex]\[
\text{Probability} = \frac{70}{1001}
\][/tex]
5. Simplify the result if necessary:
- In this case, [tex]\(\frac{70}{1001}\)[/tex] is already in simplest form.
- Approximating, we find the probability to be about 0.06993006993006994.
Given these calculations, the answer you're looking for matches with [tex]\( \frac{70}{1001} \)[/tex], which simplifies directly to the provided choice:
[tex]\[
\boxed{\frac{10}{143}}
\][/tex]
