Question

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?

Answer 1

Answer:

The probability that the committee consists of all boys is:

                       $\dfrac{10}{143}$

Step-by-step explanation:

Since a committee of four people is to be formed from a group of:

           8 boys and 6 girls.

This implies total number of people= 14.

Probability that the committee consist of all boys is calculated by taking the ratio of probability of selecting 4 boys from all boys and selecting 4 person from total number of person.

Hence, the probability is calculated by:

$\text{Probability}=\dfrac{8_C_4}{14_C_4}$

Now,

$8_C_4=\dfrac{8!}{4!\times (8-4)!}\\\\\\8_C_4=\dfrac{8!}{4!\times 4!}$

and

${14}_C_4=\dfrac{14!}{4!\times (14-4)!}\\\\\\14_C_4=\dfrac{14!}{4!\times 10!}$

Hence, the probability is given by:

$\text{Probability}=\dfrac{10}{143}$

  Hence, the probability is:

                    $\dfrac{10}{143}$

Answer 2

$\displaystyle |\Omega|=\binom{14}{4}=\dfrac{14!}{4!10!}=\dfrac{11\cdot12\cdot13\cdot14}{2\cdot3\cdot4}=1001\\ |A|=\binom{8}{4}=\dfrac{8!}{4!4!}=\dfrac{5\cdot6\cdot7\cdot8}{2\cdot3\cdot4}=70\\\\ P(A)=\dfrac{70}{1001}=\dfrac{10}{143}\approx7\%$