To find the probability of selecting a committee with a certain combination of party members, we can use the formula for combinations:
P(A) = n! / (r! * (n-r)!)
Where:
-P(A) is the probability of event A occurring
-n is the total number of items (in this case, the total number of council members)
-r is the number of items we are choosing (in this case, the number of members on the committee)
Using this formula, we can find the probability of each event as follows:
-All Republicans: There are 4 Republicans on the council, and we are selecting 3 of them for the committee. Therefore, the probability is:
P(A) = 10! / (3! * 7!) = 120 / (6 * 5040) = 1/42 = 0.02381
-All Democrats: There are 3 Democrats on the council, and we are selecting 3 of them for the committee. Therefore, the probability is:
P(A) = 10! / (3! * 7!) = 120 / (6 * 5040) = 1/42 = 0.02381
-One of each party: There are 4 Republicans, 3 Democrats, and 3 Independents on the council, and we are selecting 1 of each for the committee. There are a total of 10 * 9 * 8 = 720 ways to select 1 member from each party. Therefore, the probability is:
P(A) = 720 / (10 * 9 * 8) = 1/56 = 0.017857
-Two Democrats and one Independent: There are 3 Democrats and 3 Independents on the council, and we are selecting 2 Democrats and 1 Independent for the committee. There are a total of 3 * 2 * 3 = 18 ways to select these members. Therefore, the probability is:
P(A) = 18 / (10 * 9 * 8) = 1/280 = 0.003571
-One Independent and two Republicans: There are 4 Republicans and 3 Independents on the council, and we are selecting 2 Republicans and 1 Independent for the committee. There are a total of 4 * 3 * 3 = 36 ways to select these members. Therefore, the probability is:
P(A) = 36 / (10 * 9 * 8) = 1/140 = 0.007143
Thus, the probabilities of selecting a committee with the specified combinations of party members are:
1-All Republicans: 0.02381
2-All Democrats: 0.02381
3-One of each party: 0.017857
4-Two Democrats and one Independent: 0.003571
5-One Independent and two Republicans: 0.007143