A town council has 11 members, 6 Democrats and 5 Republicans.
If a 3‐person committee is selected at random, what is the probability that
Republicans make up the majority?
To find the probability that Republicans make up the majority in a 3-person committee randomly selected from a town council with 6 Democrats and 5 Republicans, we need to calculate the number of favorable outcomes and the total number of possible outcomes.
First, let's consider the possible scenarios where Republicans make up the majority:
1. Selecting 2 Republicans and 1 Democrat: There are 5 ways to choose 2 Republicans out of 5 and 6 ways to choose 1 Democrat out of 6, resulting in a total of 5 * 6 = 30 favorable outcomes.
2. Selecting 3 Republicans: There is only 1 way to choose all 3 Republicans.
The total number of possible outcomes can be calculated by selecting any 3 members out of the 11 council members, which is denoted as "11 choose 3" or written as 11C3.
Using the binomial coefficient formula, "n choose k" = n! / (k!(n-k)!), we can calculate 11C3:
11C3 = 11! / (3!(11-3)!) = (11 * 10 * 9) / (3 * 2) = 165.
Therefore, the total number of possible outcomes is 165.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes = (30 + 1) / 165 = 31 / 165 ≈ 0.1879.
So, the probability that Republicans make up the majority in a randomly selected 3-person committee is approximately 0.1879 or 18.79%.
To find the probability that Republicans make up the majority in a 3-person committee randomly selected from a town council with 11 members (6 Democrats and 5 Republicans), we need to determine the number of favorable outcomes and the total number of possible outcomes.
First, let's calculate the total number of possible outcomes. To form a committee of 3 people from 11 members, we use combinations, denoted as C(n, r), where n is the number of objects to choose from and r is the number of objects to choose. In this case, it is C(11, 3) since we are choosing a committee of 3 members from a council of 11. Combinations are calculated as:
C(11, 3) = 11! / (3! * (11 - 3)!) = 165
So, there are 165 possible ways to form a committee of 3 people from the council.
Next, let's calculate the number of favorable outcomes. For Republicans to make up the majority, we need at least 2 Republicans in the committee. There are two possible scenarios:
1. Selecting 2 Republicans and 1 Democrat: The number of ways to choose 2 Republicans from a group of 5 is C(5, 2) = 10, and the number of ways to choose 1 Democrat from 6 is C(6, 1) = 6. Therefore, there are 10 * 6 = 60 favorable outcomes for this scenario.
2. Selecting all 3 Republicans: There is only 1 way to select all 3 Republicans, so there is 1 favorable outcome for this scenario.
Adding up the favorable outcomes from both scenarios, we have 60 + 1 = 61 favorable outcomes.
Now, we can calculate the probability by dividing the number of favorable outcomes (61) by the total number of possible outcomes (165):
Probability = Favorable outcomes / Total outcomes
Probability = 61 / 165 ≈ 0.3697
Therefore, the probability that Republicans make up the majority in a 3-person committee randomly selected from the town council is approximately 0.3697 or 36.97%.
So you want 2 R's and 1 D
prob = C(5,2)xC(6,1)/C(11,3)
= 10(6)/165
= 4/11