To solve the problem of determining the probability that a four-person committee chosen randomly from a group consisting of eight boys and six girls will consist entirely of boys, follow these steps:
1. Determine the total number of people:
- There are 8 boys and 6 girls, making a total of [tex]\( 8 + 6 = 14 \)[/tex] people.
2. Calculate the total number of ways to choose 4 people out of 14:
- The number of ways to choose 4 people out of 14 can be calculated using the combination formula [tex]\( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)[/tex], where [tex]\( n \)[/tex] is the total number of items to choose from, and [tex]\( k \)[/tex] is the number of items to choose.
- Hence, the total number of ways to choose 4 out of 14 is [tex]\( \binom{14}{4} \)[/tex].
3. Calculate the number of ways to choose 4 boys out of 8:
- Similarly, the number of ways to choose 4 boys out of 8 is [tex]\( \binom{8}{4} \)[/tex].
4. Calculate the probability:
- The probability that the committee consists entirely of boys is the ratio of the number of ways to choose 4 boys to the total number of ways to choose any 4 people.
- Therefore, the probability is given by [tex]\( \frac{\binom{8}{4}}{\binom{14}{4}} \)[/tex].
5. Evaluate the combinations:
- Evaluating [tex]\( \binom{8}{4} \)[/tex] gives 70.
- Evaluating [tex]\( \binom{14}{4} \)[/tex] gives 1001.
6. Compute the probability:
- The probability [tex]\( P \)[/tex] that the committee consists of all boys is then [tex]\( \frac{\binom{8}{4}}{\binom{14}{4}} = \frac{70}{1001} \)[/tex].
7. Simplify the fraction if possible:
- The fraction [tex]\( \frac{70}{1001} \)[/tex] is already in its simplest form.
8. Match the result to the given choices:
- Comparing [tex]\( \frac{70}{1001} \)[/tex] to the given options, [tex]\( \frac{70}{1001} \)[/tex] can be rewritten as [tex]\( \frac{70 \times 1}{1001} = \frac{70}{1001} \)[/tex].
So the closest fraction from the given options is:
[tex]\[ \frac{70}{1001} \][/tex]
Therefore, the probability that the committee consists entirely of boys is [tex]\( \boxed{\frac{70}{1001}} \)[/tex].
