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.


a. What is the probability that the committee is composed of all Democrats?"

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 Democrats. Since there are 55 Democrats, calculate \(\binom{55}{7}\).
4. Calculate the probability that the committee is all Democrats by dividing the number of favorable outcomes by the total number of possible outcomes: \(P = \frac{\binom{55}{7}}{\binom{100}{7}}\).
5. Interpret the result as the likelihood that a randomly selected committee of 7 senators consists only of Democrats.

Key Concepts

Combinatorics and Counting Principles

Combinatorics involves counting the number of ways to select items from a set without regard to order. In this problem, combinations are used to calculate how many different committees of seven senators can be formed from the total Senate members or from a subgroup like Democrats.

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 probability that the committee consists entirely of Democrats is the number of all-Democrat committees divided by the total number of possible 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 the selections are dependent, which is important when calculating combinations and probabilities.