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, 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:
In a 3-person committee, there are C(11,3) ways to select any 3 members from a total of 11 members. The formula for combinations is given by C(n,r) = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be selected. Plugging in the values, C(11,3) = 11! / (3!(11-3)!) = 165.
Next, we need to determine the number of favorable outcomes where Republicans make up the majority. This can happen in two ways: either all 3 members are Republicans (5C3) or 2 Republicans and 1 Democrat (5C2 * 6C1). The formula for combinations is used again, where 5C3 = 5! / (3!(5-3)!) = 10 and 5C2 * 6C1 = (5! / (2!(5-2)!) * 6! / (1!(6-1)!)) = (10 * 6) = 60.
Therefore, the total number of favorable outcomes is 10 + 60 = 70.
Finally, we can determine the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes
Probability = 70 / 165 = 14 / 33.
Hence, the probability that Republicans make up the majority is 14/33 or approximately 0.4242.