A committe of 5 members will be chosen from a group of 10 teachers and 5 students. What is the probability that the committee
will have
A) all teachers?
B) 3 teachers and 2 students?
C) 3 or 4 teachers?
Number of possible choices of 5 members from a total of 15
= C(15,5) = 3003
a) number of all teachers = C(10,5) = 252
so prob(5 teachers) = 252/3003 = 12/143
b) 3 teachers, and 2 students
prob = C(10,3) * C(5,2)/3003 = ....
c) not clear with the wording.
Is it exactly 3 teachers or exactly 4 teachers, or is the case of 3 teachers included in 4 teachers?
I will go with the "exactly".
prob = ( C(10,3)C(5,2) + C(10,4)C(5,1) )/3003 =
To find the probability for each case, we need to determine the total number of possible committees and the number of favorable outcomes in each case.
A) Probability of all teachers:
To calculate this probability, we need to determine the total number of possible committees consisting of 5 members selected from the total group of 10 teachers and 5 students.
The total number of possible committees can be calculated using the combination formula:
nCr = n! / r! * (n-r)!
In this case, we want to choose 5 members from a group of 10 teachers, so the formula becomes:
10C5 = 10! / 5! * (10-5)!
The number of favorable outcomes is 1, as we want all members to be teachers.
Therefore, the probability of all teachers in the committee is:
P(all teachers) = favorable outcomes / total possible outcomes
P(all teachers) = 1 / 10C5
B) Probability of 3 teachers and 2 students:
To calculate this probability, we again need to determine the total number of possible committees consisting of 5 members selected from the total group of 10 teachers and 5 students, and then find the number of favorable outcomes where there are 3 teachers and 2 students.
The total number of possible committees is the same as in part A.
To find the number of favorable outcomes, we need to calculate two separate combinations:
- Choose 3 teachers from 10 teachers: 10C3
- Choose 2 students from 5 students: 5C2
The number of favorable outcomes is the product of these two combinations.
Therefore, the probability of 3 teachers and 2 students in the committee is:
P(3 teachers and 2 students) = favorable outcomes / total possible outcomes
P(3 teachers and 2 students) = (10C3 * 5C2) / 10C5
C) Probability of 3 or 4 teachers:
To calculate this probability, we need to determine the total number of possible committees consisting of 5 members selected from the total group of 10 teachers and 5 students, and then find the number of favorable outcomes where there are either 3 or 4 teachers.
The total number of possible committees is the same as in part A.
To find the number of favorable outcomes, we need to calculate two separate combinations:
- Choose 3 teachers from 10 teachers: 10C3
- Choose 2 students from 5 students: 5C2
- Choose 4 teachers from 10 teachers: 10C4
- Choose 1 student from 5 students: 5C1
The number of favorable outcomes is the sum of these two combinations.
Therefore, the probability of having 3 or 4 teachers in the committee is:
P(3 or 4 teachers) = favorable outcomes / total possible outcomes
P(3 or 4 teachers) = (10C3 * 5C2 + 10C4 * 5C1) / 10C5