A COMMITTEE OF 5 PEOPLE IS TO BE FORMED FROM 15 PARENTS AND 10 teachers. Find the probability that the committee will consist of at least one parent.
The chance that there are no parents is
10/25 * 9/24 * 8/23 * 7/22 * 6/21 = 6/1265
So, the chance that there is at least one parent is 1259/1265
17
To find the probability that the committee will consist of at least one parent, we can use the concept of complementary probability.
First, let's find the probability that the committee does not have any parents. To do this, we need to select all 5 members of the committee from the group of teachers only.
The number of ways to select 5 members from the 10 teachers is given by the combination formula:
10C5 = 10! / (5!(10-5)!) = 252.
Next, let's find the total number of ways to select a committee of 5 members from the combined group of parents and teachers.
The number of ways to select 5 members from the total group of 15 parents and 10 teachers is given by the combination formula:
(15 + 10)C5 = 25C5 = 25! / (5!(25-5)!) = 53,130.
Now, we can calculate the probability of selecting a committee with no parents:
P(No parents) = Number of ways to select a committee with no parents / Total number of ways to select a committee
P(No parents) = 252 / 53,130 ≈ 0.0047
Since we are interested in the probability of at least one parent, we can subtract the probability of having no parents from 1:
P(At least one parent) = 1 - P(No parents)
P(At least one parent) = 1 - 0.0047
P(At least one parent) ≈ 0.9953
Therefore, the probability that the committee will consist of at least one parent is approximately 0.9953, or 99.53%.
To find the probability that the committee will consist of at least one parent, we first need to determine the total number of possible committees that can be formed from the available individuals.
We have 15 parents and 10 teachers, so the total number of individuals to choose from is 15 + 10 = 25.
Now, let's consider the number of ways to form a committee without any restrictions. We can choose any subset of 5 people from the 25 available. This can be calculated using the combination formula:
C(n, k) = n! / (k! * (n-k)!)
where C(n, k) represents the number of ways to choose k items from a set of n items.
In this case, we want to find the number of ways to choose 5 people from 25:
C(25, 5) = 25! / (5! * (25-5)!)
= 25! / (5! * 20!)
= (25*24*23*22*21) / (5*4*3*2*1)
= 53,130
So, there are a total of 53,130 possible committees that can be formed from the 25 individuals.
Next, we need to determine the number of committees that consist of at least one parent. To do this, we can calculate the number of committees that have no parents and subtract this from the total number of committees.
To form a committee with no parents, we can only choose from the 10 teachers. So, the number of ways to form a committee without any parents is:
C(10, 5) = 10! / (5! * (10-5)!)
= 10! / (5! * 5!)
= (10*9*8*7*6)/(5*4*3*2*1)
= 252
Therefore, there are 252 committees that consist of only teachers.
Finally, we can calculate the number of committees that consist of at least one parent by subtracting the number of committees with no parents from the total number of committees:
Total number of committees with at least one parent = Total number of committees - Number of committees with no parents
= 53,130 - 252
= 52,878
So, there are 52,878 committees that consist of at least one parent.
To find the probability, we divide the number of favorable outcomes (committees with at least one parent) by the total number of possible outcomes (all committees):
Probability = Number of favorable outcomes / Total number of possible outcomes
= 52,878 / 53,130
= 0.9952 (rounded to 4 decimal places)
Therefore, the probability that the committee will consist of at least one parent is approximately 0.9952.