A committee is to be formed to investigate what activities teenagers have available in small communities. The committee is to have 7 members, chosen randomly from interested community members. There are 10 parents, 5 teenagers, and 4 adults without children who have all expressed an interest in serving on the committee.

Question

A committee is to be formed to investigate what activities teenagers have available in small communities. The committee is to have 7 members, chosen randomly from interested community members. There are 10 parents, 5 teenagers, and 4 adults without children who have all expressed an interest in serving on the committee.

What is the probability that the adults without children are all on the committee? Express your answer to the nearest tenth of one percent.

Answer 1

number of possible committees without restrictions

= C(19,7) = 50388

number of committees with the specified 4 adults
= 1 x C(15, 3) , (we must still choose 3 of the remaining 15)= 455

prob of your event = 455/50388 = 35/3876
= .0090
= appr .9%

Answer 2

To find the probability that the adults without children are all on the committee, we need to find the total number of possible committees and the number of committees where the adults without children are all included.

First, let's determine the number of possible committees. Since there are a total of 19 interested community members (10 parents, 5 teenagers, and 4 adults without children), we can choose 7 members from these 19. We can use the combination formula to calculate this:

C(19, 7) = 19! / (7!(19 - 7)!) = 19! / (7!12!) = (19 * 18 * 17 * 16 * 15 * 14 * 13) / (7 * 6 * 5 * 4 * 3 * 2 * 1) = 54,264

Now, let's determine the number of committees where the adults without children are all included. Since there are 4 adults without children, we need to choose all 4 of them and then choose the remaining 3 members from the remaining 15 interested community members (10 parents and 5 teenagers). Again, we can use the combination formula to calculate this:

C(15, 3) = 15! / (3!(15 - 3)!) = 15! / (3!12!) = (15 * 14 * 13) / (3 * 2 * 1) = 455

Therefore, the probability that the adults without children are all on the committee is:

P = (number of committees with all adults without children) / (total number of possible committees) = 455 / 54,264 ≈ 0.0084

To express this probability as a percentage, we can multiply by 100:

Percentage = 0.0084 * 100 ≈ 0.84%

Therefore, the probability that the adults without children are all on the committee is approximately 0.84%.