5. Congressional Committee Memberships The
composition of the 108th Congress is 51 Republicans,
48 Democrats, and 1 Independent. A committee on aid
to higher education is to be formed with 3 Senators to
be chosen at random to head the committee. Find the
probability that the group of 3 consists of
a. All Republicans
b. All Democrats
c. One Democrat, one Republican, and one
Independent
The probability of all events occurring is found by multiplying the probability of the individual events.
a. (51/100)(50/99)(49/98) = ?
b. Work similar to a.
c. (48/100)(51/99)(1/98) = ?
To find the probability of each scenario, we need to calculate the number of favorable outcomes and divide it by the total number of possible outcomes.
a. Probability of choosing all Republicans:
In this case, we will choose 3 Senators from the group of 51 Republicans. The total number of possible outcomes is choosing 3 Senators from the total 100 Senators (51 Republicans + 48 Democrats + 1 Independent).
Number of favorable outcomes: 51C3 (combining 51 Republicans into groups of 3)
Total number of possible outcomes: 100C3 (choosing 3 Senators out of 100)
P(all Republicans) = 51C3 / 100C3
b. Probability of choosing all Democrats:
In this case, we will choose 3 Senators from the group of 48 Democrats. The total number of possible outcomes is still choosing 3 Senators from the total 100 Senators.
Number of favorable outcomes: 48C3 (combining 48 Democrats into groups of 3)
Total number of possible outcomes: 100C3 (choosing 3 Senators out of 100)
P(all Democrats) = 48C3 / 100C3
c. Probability of choosing one Democrat, one Republican, and one Independent:
In this case, we will choose 1 Senator from each group: Democrats, Republicans, and Independents. The total number of possible outcomes remains choosing 3 Senators from the total 100 Senators.
Number of favorable outcomes: 48C1 * 51C1 * 1C1 (choosing 1 Senator from each group)
Total number of possible outcomes: 100C3 (choosing 3 Senators out of 100)
P(1 Democrat, 1 Republican, 1 Independent) = (48C1 * 51C1 * 1C1) / 100C3
Note: 'nCk' represents the number of combinations of n items taken k at a time, calculated as n! / (k! * (n-k)!)
Evaluate these expressions to get the final probabilities.
To find the probability, we need to calculate the number of favorable outcomes (groups of 3 Senators) for each scenario and divide it by the total number of possible outcomes.
a. All Republicans:
There are 51 Republicans in the Congress, so we need to choose 3 of them. The number of ways to choose 3 Republicans out of 51 is calculated using the combination formula:
C(51, 3) = 51! / (3!(51-3)!) = 51! / (3!48!) = (51 * 50 * 49) / (3 * 2 * 1) = 22,825
Now, we need to calculate the total number of possible outcomes, which is choosing any 3 Senators out of the total 100 Senators:
C(100, 3) = 100! / (3!(100-3)!) = 100! / (3!97!) = (100 * 99 * 98) / (3 * 2 * 1) = 1,960,200
Therefore, the probability of choosing a group of 3 Senators consisting of all Republicans is:
P(All Republicans) = favorable outcomes / total outcomes = 22,825 / 1,960,200 ≈ 0.0117
So, the probability is approximately 0.0117 or 1.17%.
b. All Democrats:
Following the same logic as above, the number of ways to choose 3 Democrats out of the 48 Democrats is:
C(48, 3) = 48! / (3!(48-3)!) = 48! / (3!45!) = (48 * 47 * 46) / (3 * 2 * 1) = 17,296
The total number of possible outcomes remains the same as before:
C(100, 3) = 1,960,200
Therefore, the probability of choosing a group of 3 Senators consisting of all Democrats is:
P(All Democrats) = favorable outcomes / total outcomes = 17,296 / 1,960,200 ≈ 0.0088
So, the probability is approximately 0.0088 or 0.88%.
c. One Democrat, one Republican, and one Independent:
In this scenario, we need to choose 1 Senator from each party. The number of ways to choose 1 Democrat out of the 48 Democrats is:
C(48, 1) = 48
The number of ways to choose 1 Republican out of the 51 Republicans is:
C(51, 1) = 51
The number of ways to choose 1 Independent out of the 1 Independent is:
C(1, 1) = 1
Now, we need to calculate the total number of possible outcomes, which is still choosing any 3 Senators out of the total 100 Senators:
C(100, 3) = 1,960,200
Therefore, the probability of choosing a group of 3 Senators consisting of one Democrat, one Republican, and one Independent is:
P(One Democrat, One Republican, One Independent) = (favorable outcomes) / (total outcomes)
= (48 * 51 * 1) / 1,960,200 ≈ 0.0012
So, the probability is approximately 0.0012 or 0.12%.