There are 8 speaking roles in the play. Of the 8 speaking roles, 5 are for males and 3 are for females. After the 8 speaking roles in the play are cast, the remaining cast members will make up the chorus.

C. What is the probability that the set of siblings consisting of a brother and his 2 sisters are all in the chorus?

My work ( what I did) :

5/8 ... 3/8 so wouldn't you subject the 3 from 5? I am confused

To find the probability that the set of siblings consisting of a brother and his 2 sisters are all in the chorus, we need to consider the number of ways the siblings can be in the chorus and divide it by the total number of possible casts for the chorus.

First, let's determine the number of ways the brother and his 2 sisters can be in the chorus. Since there are 5 male roles and 3 female roles, the brother has a 5/8 chance of being cast in a male role. After the brother is cast, there are 2 female roles remaining, from which the first sister has a 2/7 chance of being cast. Finally, after the first sister is cast, there is 1 female role remaining for the second sister, giving her a 1/6 chance of being cast. Therefore, the probability of the set of siblings all being in the chorus is (5/8) * (2/7) * (1/6).

However, we need to account for the fact that there are still cast members to be selected for the remaining roles in the chorus. The number of cast members who will make up the chorus is given by 8 (total roles) - 3 (speaking roles) = 5. The remaining cast members can be chosen from the remaining 8 - 3 = 5 roles without any restrictions, so the number of possible casts for the chorus is 5!/5! (5 factorial divided by 5 factorial) = 1.

Therefore, the probability that the set of siblings consisting of a brother and his 2 sisters are all in the chorus is (5/8) * (2/7) * (1/6) * 1 = 10/336 = 5/168.

To find the probability that the set of siblings consisting of a brother and his 2 sisters are all in the chorus, we need to consider the total number of possible casts and the number of casts that satisfy the condition.

Since there are 8 speaking roles in total, we can choose 3 individuals from the remaining cast members (as the brother and his 2 sisters are already assigned speaking roles) to make up the chorus. The number of ways to select 3 individuals from the remaining cast members is given by the combination formula, denoted as "nCr".

The formula for nCr is:
nCr = n! / [(n-r)! * r!]

Here, n represents the total number of cast members and r represents the number of individuals we want to choose.

In this scenario, n is equal to the remaining cast members after the speaking roles are cast, which is 8 - 1 - 2 = 5 (as the brother and his 2 sisters are already assigned roles). And r is equal to the number of individuals we want to select, which is 3.

So, the number of ways to select 3 individuals to form the chorus is:
5C3 = 5! / [(5-3)! * 3!]
= 5! / (2! * 3!)
= (5 * 4 * 3!) / (2! * 3!)
= (5 * 4) / 2
= 10

Now, we need to determine the total number of possible casts, which is the total number of ways to assign speaking roles to all 8 cast members. This is calculated using the permutation formula, denoted as "nPr".

The formula for nPr is:
nPr = n! / (n-r)!

Here, n is equal to the total number of cast members, which is 8. And r is equal to the number of speaking roles, which is also 8.

So, the total number of possible casts is:
8P8 = 8! / (8-8)!
= 8! / 0!
= 8! / 1
= 8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
= 40,320

To find the probability, we divide the number of favorable outcomes (the number of casts where the siblings are all in the chorus) by the total number of possible casts.

Therefore, the probability is:
Probability = Number of favorable outcomes / Total number of possible casts
= 10 / 40,320

Simplifying the fraction, we get:
Probability = 1 / 4,032

So, the probability that the set of siblings consisting of a brother and his 2 sisters are all in the chorus is 1 in 4,032.