Suppose that a state university has to form a committee of 5 members from a list of 20 candidates out of whom 12 are teachers and 8 are students.If members of the committee are selected at random, what is the probability that the majority of the committee members are students?
To find the probability that the majority of the committee members are students, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
Step 1: Calculating the number of favorable outcomes
In this case, the majority of the committee members are students. This means that there will be 3 or 4 student members on the committee. We need to calculate the number of different ways this can happen.
Let's consider the two cases separately:
Case 1: 3 student members, 2 teacher members
The number of ways to choose 3 students out of 8 is given by the combination formula: C(8,3) = 8! / (3! * (8-3)!) = 56
Similarly, the number of ways to choose 2 teachers out of 12 is given by: C(12,2) = 12! / (2! * (12-2)!) = 66
Therefore, the total number of ways to form a committee with 3 student members and 2 teacher members is 56 * 66 = 3,696.
Case 2: 4 student members, 1 teacher member
The number of ways to choose 4 students out of 8 is given by: C(8,4) = 8! / (4! * (8-4)!) = 70
The number of ways to choose 1 teacher out of 12 is given by: C(12,1) = 12! / (1! * (12-1)!) = 12
Therefore, the total number of ways to form a committee with 4 student members and 1 teacher member is 70 * 12 = 840.
So, the total number of favorable outcomes is 3,696 + 840 = 4,536.
Step 2: Calculating the total number of possible outcomes
To calculate the total number of possible outcomes, we need to choose 5 committee members out of 20 candidates. This can be calculated using the combination formula: C(20,5) = 20! / (5! * (20-5)!) = 15,504.
Step 3: Calculating the probability
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of Favorable Outcomes / Total Number of Possible Outcomes
Probability = 4,536 / 15,504
Probability ≈ 0.2928 or 29.28%
Therefore, the probability that the majority of the committee members are students is approximately 29.28%.