A selection committee consisting of 10 members is to be formed from a group of 20 employees, 12 are female. calculate the probability that that at most 8 committee members will be female
To calculate the probability that at most 8 committee members will be female, we need to consider two scenarios: when there are 8 female members and when there are fewer than 8 female members.
First, let's calculate the probability of selecting exactly 8 female members:
- The number of ways to select 8 members out of 12 female employees is given by the combination formula: C(12, 8) = (12! / (8! * (12-8)!)) = 495.
- The remaining 2 members need to be selected from the 8 male employees from a total of 20 - 12 = 8 male employees. The number of ways to select 2 members out of 8 male employees is given by the combination formula: C(8, 2) = (8! / (2! * (8-2)!)) = 28.
- Therefore, the total number of ways to select a committee with exactly 8 female members is 495 * 28 = 13,860.
Next, let's calculate the probability of selecting fewer than 8 female members:
- For each possible number of female members from 0 to 7, we will calculate the number of ways to select committee members with that number of females and sum them up.
- If there are x female members, the number of ways to select x female members from the 12 available female employees is C(12, x). Similarly, the number of ways to select (10-x) male members from the 8 available male employees is C(8, 10-x).
- For each value of x from 0 to 7, multiply the number of ways to select x female members by the number of ways to select (10-x) male members, and sum up all the values.
- Therefore, the total number of ways to select a committee with fewer than 8 female members is given by the sum of (C(12, x) * C(8, 10-x)) for x = 0 to 7.
Finally, we need to calculate the total number of ways to select a committee with 10 members from a group of 20 employees:
- The total number of ways to select 10 members out of 20 is given by the combination formula: C(20, 10) = (20! / (10! * (20-10)!)) = 184,756.
Now we can calculate the probability:
- The probability of selecting a committee with at most 8 female members is given by (number of favorable outcomes) / (total number of possible outcomes).
- The number of favorable outcomes is the sum of the number of ways to select exactly 8 female members and the number of ways to select fewer than 8 female members.
- Therefore, the probability is (13,860 + sum of (C(12, x) * C(8, 10-x)) for x = 0 to 7) / 184,756.