Selecting a Committee
Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly.
b. What is the probability that the committee is composed of all Republicans?

Verified step by step guidance

1. Identify the total number of senators, which is the sum of Democrats and Republicans: \$55 + 45 = 100$ senators.
2. Determine the total number of ways to select a committee of 7 senators from the 100 senators. This is a combination problem, so use the formula for combinations: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here, calculate \(\binom{100}{7}\).
3. Determine the number of ways to select a committee of 7 senators composed entirely of Republicans. Since there are 45 Republicans, calculate \(\binom{45}{7}\).
4. Calculate the probability that the committee is all Republicans by dividing the number of all-Republican committees by the total number of possible committees: \(P = \frac{\binom{45}{7}}{\binom{100}{7}}\).
5. Interpret the result as the likelihood of randomly selecting a 7-member committee consisting only of Republicans from the entire Senate.

Key Concepts

Combinatorics and Combinations

Combinatorics involves counting the number of ways to select items from a set without regard to order. Combinations specifically count the number of ways to choose k items from n distinct items, calculated as n! / (k!(n-k)!). This is essential for determining the total number of possible committees.

Probability of an Event

Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. Here, the event is selecting a committee of all Republicans, so the probability is the number of all-Republican committees divided by the total number of committees.

Random Sampling Without Replacement

Random sampling without replacement means selecting members such that each chosen senator cannot be selected again. This affects the total number of possible committees and ensures that combinations rather than permutations are used, as order does not matter.