A four-person committee is chosen from a group of eight boys and six girls. If students are chosen at random, what is the probability that the committee consists of all boys?
$\frac{4}{1001}$
$\frac{15}{1001}$
$\frac{10}{143}$
$\frac{133}{143}$
Answer (1)
Calculate the total number of ways to form a 4-person committee from 14 students: ( 4 14 ) = 1001 .
Calculate the number of ways to form a 4-boy committee from 8 boys: ( 4 8 ) = 70 .
Divide the number of all-boy committees by the total number of committees to find the probability: 1001 70 .
Simplify the fraction to get the final probability: 143 10 .
Explanation
Understand the problem We have a group of 8 boys and 6 girls, making a total of 14 students. We want to form a four-person committee chosen at random from these 14 students. Our goal is to find the probability that the committee consists of all boys.
Calculate the total number of possible committees First, we need to calculate the total number of ways to choose a four-person committee from the 14 students. This is a combination problem, since the order in which the students are chosen does not matter. The total number of ways to choose 4 students from 14 is given by the combination formula: ( 4 14 ) = 4 ! ( 14 − 4 )! 14 ! = 4 ! 10 ! 14 ! Calculating this value: ( 4 14 ) = 4 × 3 × 2 × 1 14 × 13 × 12 × 11 = 14 × 13 × 4 × 3 × 2 12 × 11 = 14 × 13 × 2 1 × 11 = 7 × 13 × 11 = 1001 So there are 1001 possible four-person committees.
Calculate the number of all-boys committees Next, we need to calculate the number of ways to choose a committee consisting of all boys. Since there are 8 boys, we want to choose 4 of them. This is also a combination problem, and the number of ways to choose 4 boys from 8 is given by: ( 4 8 ) = 4 ! ( 8 − 4 )! 8 ! = 4 ! 4 ! 8 ! Calculating this value: ( 4 8 ) = 4 × 3 × 2 × 1 8 × 7 × 6 × 5 = 2 × 7 × 3 × 2 6 × 5 = 2 × 7 × 1 × 5 = 70 So there are 70 ways to choose a committee of 4 boys from the 8 boys.
Calculate the probability and simplify Now, we can calculate the probability that the committee consists of all boys. This is the number of ways to choose an all-boys committee divided by the total number of possible committees: P ( all boys ) = ( 4 14 ) ( 4 8 ) = 1001 70 We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 7: 1001 70 = 1001 ÷ 7 70 ÷ 7 = 143 10 Thus, the probability that the committee consists of all boys is 143 10 .
Examples
This type of probability calculation is useful in scenarios like forming teams or selecting groups where you want to know the likelihood of a specific demographic makeup. For instance, if a company is randomly selecting a team for a project from a pool of employees with varying skill sets, they might want to calculate the probability of the team having a certain number of members with specific expertise. This helps in understanding the diversity and skill distribution within randomly selected groups.
