Problem 37 - Algebra for College Students

Question

Suppose that a five-person committee is selected at random from the eight people \(\mathrm{Al}\), Barb, Chad, Dominique, Eric, Fern, George, and Harriet. Find the probability of each of the following events:Either Chad or Dominique, but not both, is on the committee. \(\frac{15}{28}\)

Step-by-Step Solution

Answer

The probability is \( \frac{15}{28} \).

Step 1: Calculate Total Possible Committees

The first step is to calculate how many total 5-person committees can be formed from the 8 people. This can be calculated using combinations: \( \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5!3!} = 56 \). So, there are 56 possible committees.

Step 2: Calculate Committees with Chad but not Dominique

Next, we calculate how many committees include Chad but exclude Dominique. We first select Chad, then choose 4 more members from the remaining 6 people (since Dominique is not included). Use combinations: \( \binom{6}{4} = 15 \).

Step 3: Calculate Committees with Dominique but not Chad

Similarly, calculate how many committees include Dominique but exclude Chad. We first select Dominique, then choose 4 more members from the remaining 6 people. Use combinations: \( \binom{6}{4} = 15 \).

Step 4: Total Committees with Either Chad or Dominique but not Both

To find the total number of committees with either Chad or Dominique but not both, add the results from Steps 2 and 3: \( 15 + 15 = 30 \).

Step 5: Calculate Probability

Finally, compute the probability by dividing the favorable outcomes by the total outcomes: \( \frac{30}{56} = \frac{15}{28} \).

Key Concepts

Combinations

Combinations are a fundamental concept in probability and discrete mathematics. They are used to calculate how many ways we can select items from a larger group, where the order of selection doesn't matter. In the given exercise, the use of combinations is essential in determining the possible committees that can be chosen from a group of eight people.

• Combinations take into account the number of ways to pick a subset without regard to the order in which they are picked.
• Mathematically, the number of combinations of selecting \( r \) objects from \( n \) objects is denoted as \( \binom{n}{r} \), and it can be calculated using the formula:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]Here, "!" denotes factorial, meaning the product of all positive integers up to that number.
In this exercise, we needed combinations to find out how to create a 5-person committee from 8 people. Calculating \( \binom{8}{5} \) helped us find the total number of committee configurations possible.

Discrete Mathematics

Discrete mathematics deals with distinct and separate values, unlike continuous mathematics that involves calculus and analysis. This branch plays a significant role in computer science, logic, and set theory, often focusing on countable structures.

• In the context of this exercise, discrete mathematics helps by providing methods to work with finite sets of people while calculating probabilities.
• It involves working with whole numbers and making logical deductions, which are key when determining the combinations of people for the committee.

Discrete mathematics enables us to apply combinatorial techniques, essential for solving our committee-selection problem by logically analyzing different possible cases throughout the step-by-step solution.

Problem Solving

Problem-solving in mathematics involves a methodical approach to finding solutions to given problems. It often requires logical reasoning and the systematic breaking down of each problem into smaller, manageable parts.

• In the original exercise, clear steps were crucial to solving the problem of finding the probability involving Chad or Dominique in the committee.
• The steps included identifying the total possible outcomes, finding favorable outcomes where Chad or Dominique, but not both, are included, and finally computing the probability by division.

By breaking down the problem into smaller steps, it becomes easier to process complex probability questions, ensuring a deeper understanding of the concepts involved and achieving an accurate result.