To determine the probability that a randomly chosen four-person committee consists of all boys from a group of eight boys and six girls, we follow these steps:
1. Calculate the total number of ways to form a four-person committee from the 14 students (8 boys and 6 girls):
The total number of ways to choose 4 people out of 14 can be found using the combination formula:
[tex]\[
\binom{14}{4}
\][/tex]
2. Determine the number of ways to form an all-boys committee:
Since there are 8 boys, we need to calculate the number of ways to choose 4 boys out of 8:
[tex]\[
\binom{8}{4}
\][/tex]
3. Calculate the probability:
The probability that the committee consists of all boys is the ratio of the number of all-boys committees to the total number of committees:
[tex]\[
\text{Probability} = \frac{\binom{8}{4}}{\binom{14}{4}}
\][/tex]
Given the calculations:
- The total number of four-person committees from 14 students is:
[tex]\[
\binom{14}{4} = 1001
\][/tex]
- The number of four-person committees consisting entirely of boys is:
[tex]\[
\binom{8}{4} = 70
\][/tex]
Therefore, the probability that a four-person committee consists of all boys is:
[tex]\[
\text{Probability} = \frac{70}{1001}
\][/tex]
Now, let's match this fraction to one of the given options:
[tex]\[
\frac{4}{1001}, \quad \frac{15}{1001}, \quad \frac{10}{143}, \quad \frac{133}{143}
\][/tex]
The fraction [tex]\(\frac{70}{1001}\)[/tex] is equivalent to approximately 0.06993, which matches the given answer exactly.
Thus, the correct option is:
[tex]\[
\boxed{\frac{10}{143}}
\][/tex]
