A committee of five people is to be formed from six lawyers and seven teachers. Find the probability that none are lawyers.
pr=(7/13)(6/12)(5/11)(4/10)(3/9)
= 7!*913-5)!/(13!(7-5)!)
To find the probability that none of the committee members are lawyers, we need to determine the total number of ways to form a committee of five people from the six lawyers and seven teachers. Let's break down the solution step-by-step:
Step 1: Determine the total number of ways to select 5 people from the total pool of 13 individuals (6 lawyers + 7 teachers).
The total number of ways to select a committee of 5 people from a group of 13 individuals is represented by the combination "13 choose 5", denoted as C(13, 5).
C(13, 5) = 13! / (5! * 8!) = 1287
Therefore, there are 1287 possible ways to select a committee of 5 people from the group of 13 individuals.
Step 2: Determine the number of ways to select 5 people from the group of 7 teachers.
We need to select all 5 committee members from the group of 7 teachers. This can be represented by the combination "7 choose 5", denoted as C(7, 5).
C(7, 5) = 7! / (5! * 2!) = 21
Therefore, there are 21 possible ways to select a committee of 5 people consisting only of teachers.
Step 3: Determine the probability that none of the committee members are lawyers.
The probability of an event is calculated by dividing the number of favorable outcomes (none of the committee members are lawyers) by the total number of possible outcomes (selecting a committee of 5 people from the group of 13 individuals).
The number of favorable outcomes is the number of ways to select 5 committee members from the group of 7 teachers, which is 21 (determined in step 2).
The total number of possible outcomes is the number of ways to select 5 committee members from the group of 13 individuals, which is 1287 (determined in step 1).
Probability = Number of favorable outcomes / Total number of possible outcomes
Probability = 21 / 1287 = 0.0163 (rounded to four decimal places)
Therefore, the probability that none of the committee members are lawyers is approximately 0.0163 or 1.63%.
To find the probability that none of the committee members are lawyers, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
First, we determine the number of favorable outcomes, i.e., the number of committees formed entirely from teachers. Since we want none of the committee members to be lawyers, we must choose all five committee members from the group of teachers, which has seven teachers. So, we need to select 5 out of 7 teachers.
The number of ways to select 5 teachers out of 7 can be calculated using the combination formula:
C(n, r) = n! / (r!(n-r)!),
where C(n, r) represents the number of possible combinations of selecting r elements from a set of n elements.
Using this formula, the number of ways to select 5 teachers out of 7 is:
C(7, 5) = 7! / (5!(7-5)!)
= 7! / (5!2!)
= (7 × 6 × 5!)/(5! × 2)
= (7 × 6) / (2)
= 21.
Therefore, there are 21 possible committees that consist entirely of teachers.
Next, we determine the total number of possible outcomes, which is the number of committees formed from both lawyers and teachers. To do this, we need to select 5 committee members out of the total pool of 13 individuals (6 lawyers + 7 teachers).
Using the combination formula, the number of ways to select 5 committee members out of 13 is:
C(13, 5) = 13! / (5!(13-5)!)
= 13! / (5!8!)
= (13 × 12 × 11 × 10 × 9 × 8!) / (5!8!)
= (13 × 12 × 11 × 10 × 9) / (5 × 4 × 3 × 2 × 1)
= 1287.
Therefore, there are 1287 possible committees that can be formed from the group of lawyers and teachers.
Finally, we can calculate the probability that none of the committee members are lawyers by dividing the number of favorable outcomes (21) by the total number of possible outcomes (1287):
Probability = Number of favorable outcomes / Total number of possible outcomes
= 21 / 1287
≈ 0.0163 (rounded to four decimal places)
Therefore, the probability that none of the committee members are lawyers is approximately 0.0163, or 1.63%.