Problem 36 - 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:George is not on the committee. \(\frac{3}{8}\)

Step-by-Step Solution

Answer

The probability is \(\frac{3}{8}\).

Step 1: Determine Total Ways to Form Committee

Calculate the total number of ways to choose any five people from the pool of eight. Use the combination formula \(\binom{n}{r}\), where \(n\) is the total number of people and \(r\) is the number to choose. Here, \(n = 8\) and \(r = 5\). Therefore, calculate \(\binom{8}{5} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\).

Step 2: Determine Ways George is Not Included

If George is not on the committee, then the committee consists of the remaining seven people. Calculate the number of ways to choose five people from these seven people. Use the formula \(\binom{7}{5}\). Thus, \(\binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21\).

Step 3: Calculate Probability

The probability of George not being on the committee is the number of favorable outcomes (when George is not included) divided by the total outcomes. So, it's \(\frac{\text{Number of ways George is not included}}{\text{Total ways to form committee}} = \frac{21}{56}\).

Step 4: Simplify the Probability

Simplify \(\frac{21}{56}\). The greatest common divisor of 21 and 56 is 7. Therefore, \(\frac{21}{56} = \frac{21 \div 7}{56 \div 7} = \frac{3}{8}\).

Key Concepts

Combinatorics

Combinatorics is an exciting branch of mathematics focused on counting and arranging sets of objects. It explores how objects can be chosen or arranged following specific rules and constraints. In the context of our committee problem, combinatorics helps us find different ways to select members from a group without considering order. This involves using a specific type of combinatorics called "combinations," where the arrangement of the selected members does not matter. By understanding combinatorics, we can solve various problems like finding probabilities, optimizing resource allocations, and making strategic decisions in numerous fields.

Combination Formula

The combination formula is a key concept in combinatorics used to determine the number of ways to choose a subset of items from a larger set, where the order of selection does not matter. It is expressed as \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items, \( r \) is the number of items to choose, and \(!\) denotes factorial, which means multiplying a series of descending natural numbers.

For example, in our exercise, we had eight people, and we needed to select five people to form a committee. Using the combination formula, we calculated \( \binom{8}{5} \):

• First, calculate the factorial parts: \(8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).
• Then simplify using the combination formula: \( \binom{8}{5} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \).

This formula is essential for solving problems that involve selecting groups from larger pools without regard to order.

Probability Calculation

Probability calculation is the process of quantifying the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In this exercise, probability helps us determine how likely it is for George not to be part of the committee. We calculate probability by dividing the number of ways the event can occur (favorable outcomes) by the total number of possible outcomes.

To calculate the probability of George not being on the committee, we:

• Found the number of ways to form any committee: 56 ways.
• Calculated ways of excluding George: 21 ways (using combinations with the remaining 7 people).

Thus, the probability was \( \frac{21}{56} \), which simplifies to \( \frac{3}{8} \). Simplifying probabilities is essential, as it provides a clearer understanding of the event's likelihood. Overall, probability calculation is a powerful tool in various real-world applications, from insurance to game theory and more.