Problem 34 - Algebra for College Students

Question

Suppose that a committee of two boys is to be chosen at random from the five boys Al, Bill, Carl, Dan, and Eli. Find the probability of each of the following events:Dan or Eli, but not both of them, is on the committee. \(\frac{3}{5}\)

Step-by-Step Solution

Answer

The probability is \( \frac{3}{5} \).

Step 1: Understand the Problem

We are tasked with selecting a committee of 2 boys from a group of 5 boys: Al, Bill, Carl, Dan, and Eli. We need to find the probability that either Dan or Eli, but not both, is selected to be on this committee. This is an example of a combination problem in probability.

Step 2: Calculate Total Possible Combinations

First, calculate the total number of ways to choose 2 boys from the 5. This is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose.\[\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10\]Thus, there are 10 possible combinations.

Step 3: List Favorable Outcomes

List the combinations where either Dan or Eli, but not both, is included. These combinations are: - Dan and Al - Dan and Bill - Dan and Carl - Eli and Al - Eli and Bill - Eli and Carl This gives us 6 favorable outcomes where Dan or Eli, but not both, is on the committee.

Step 4: Calculate Probability

The probability of the event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes. We have 6 favorable outcomes and 10 total outcomes, so:\[P = \frac{6}{10} = \frac{3}{5}\]

Step 5: Conclude with the Simplified Probability

Therefore, the probability that either Dan or Eli, but not both, is on the committee is \( \frac{3}{5} \).

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It ranges between 0 and 1, where 0 means an event will not happen and 1 means it will certainly occur.
In our context, probability helps determine the chance of selecting a committee with specific conditions.
To calculate probability, use the formula:

• Probability = \( \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \)

The formula gives us a fraction which tells us how often we can expect the event to happen if we were to repeat the selection process many times.
For the committee example, computing the probability involves counting the ways Dan or Eli, but not both, end up in the final selections.

Combination Formula

The combination formula is used to determine the number of ways to select a subset from a larger set without regard to the order of elements.
It is represented as \( \binom{n}{r} \), where \( n \) is the total number of items, and \( r \) is the number of items to choose.
The formula is:

• \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

For the committee of boys, \( n = 5 \) and \( r = 2 \). The calculation results in 10 ways to select any 2 boys from the group without caring about order.
This total number helps us find the probability of various configurations like having either Dan or Eli, but not both, on the committee.

Favorable Outcomes

Favorable outcomes refer to situations that meet the specific criteria of the problem being solved.
In probability problems, identifying favorable outcomes helps pinpoint the specific results we desire.
For this committee selection:

• Each favorable outcome involves having either Dan or Eli on the committee, with the exclusion of the other.

The potential teams are:

• Dan and Al
• Dan and Bill
• Dan and Carl
• Eli and Al
• Eli and Bill
• Eli and Carl

This results in 6 favorable outcomes, helping determine the probability when compared to the total possible outcomes.

Event Probability

Event probability is the likelihood that a particular event occurs from a set of all possible events.
In our problem, the event is defined by Dan or Eli, but not both, being selected.
First, calculate the total number of outcomes, followed by listing all outcomes that fit the event's conditions.

• In total, there are 10 potential committees of 2 boys possible from our group of 5.
• Among these, 6 combinations meet the criteria of having either Dan or Eli, but not both.

Thus, the event probability is calculated as \( \frac{6}{10} \) or simplified to \( \frac{3}{5} \), which means out of every 5 tries, we can expect around 3 selections to satisfy the condition.