Question

There are 14 senators (half Democrats, half Republicans) and 20 representatives (half Democrats, half Republicans) who wish to serve on a joint congressional committee on tax reform. A 6-person committee is chosen at random from those who wish to serve. Complete parts A) and B) below.

(A) What is the probability that the joint committee contains equal numbers of senators and representatives?
Let the sample space S be the set of all possible 6-person committees that can be formed from those who wish to serve. The number of elements in the sample space S is n(S) =

(B) What is the probability that the joint committee contains equal numbers of Democrats and Republicans?

Answer 1

The probability that the joint committee contains equal numbers of senators and representatives is 1, while the probability that the joint committee contains equal numbers of Democrats and Republicans is approximately 0.0208.

Step-by-step explanation:

To find the probability that the joint committee contains equal numbers of senators and representatives, we need to determine the number of ways we can select 3 senators and 3 representatives from their respective groups. First, we calculate the total number of 6-person committees that can be formed from those who wish to serve. Since there are 14 senators and 20 representatives, the sample space consists of C(14,3) * C(20,3) = 1,680 committees. Next, we calculate the number of committees with equal numbers of senators and representatives. Since there are an equal number of senators and representatives, we can choose 3 senators from the available senators and 3 representatives from the available representatives. This can be calculated as C(14,3) * C(20,3) = 1,680 committees. Therefore, the probability is 1,680/1,680 = 1.

To find the probability that the joint committee contains equal numbers of Democrats and Republicans, we need to consider the party affiliation of the selected senators and representatives. Since there are 7 Democrats and 7 Republicans in each group, we need to choose 3 Democrats and 3 Republicans from their respective groups. This can be calculated as C(7,3) * C(7,3) = 35 committees. Therefore, the probability is 35/1,680 ≈ 0.0208.