Problem 19 - A Graphical Approach to College Algebra

Question

Mrs. Schmulen invites 10 relatives to a party: her mother, 2 uncles, 3 brothers, and 4 cousins. If the chances of any one guest arriving first are equally likely, find each probability. (a) The first guest is an uncle or a brother. (b) The first guest is a brother or a cousin. (c) The first guest is a brother or her mother.

Step-by-Step Solution

Answer

(a) \( \frac{1}{2} \), (b) \( \frac{7}{10} \), (c) \( \frac{2}{5} \).

Step 1: Determine the Total Number of Guests

The total number of guests invited by Mrs. Schmulen is given as 10, consisting of her mother, 2 uncles, 3 brothers, and 4 cousins. Therefore, the total number of guests is 10.

Step 2: Calculate Probability for Uncle or Brother (a)

To find the probability that the first guest is an uncle or a brother, we add the number of uncles and brothers. Mrs. Schmulen has 2 uncles and 3 brothers, which sums up to 5. The probability is then calculated by dividing the successful outcomes by the total number of outcomes: \( P(\text{Uncle or Brother}) = \frac{5}{10} = \frac{1}{2} \).

Step 3: Calculate Probability for Brother or Cousin (b)

To find the probability that the first guest is a brother or a cousin, we add the number of brothers and cousins. There are 3 brothers and 4 cousins, which totals to 7. The probability is: \( P(\text{Brother or Cousin}) = \frac{7}{10} \).

Step 4: Calculate Probability for Brother or Mother (c)

To find the probability that the first guest is a brother or her mother, we add the number of brothers and the mother. There are 3 brothers and 1 mother, which sums up to 4. The probability is: \( P(\text{Brother or Mother}) = \frac{4}{10} = \frac{2}{5} \).

Key Concepts

Event Outcome

In probability, an **event outcome** is any specific result that may occur from a chance process. In Mrs. Schmulen's party scenario, each individual invited represents a distinct event outcome. This means that every relative, such as her mother, uncles, brothers, and cousins, is a potential first arrival at the party. Calculating the probability of specific relatives arriving first involves understanding each as a separate and equally possible outcome.
To break it down:

• The first guest to arrive is an example of a single event outcome in this context.
• These outcomes are distinct and mutually exclusive. Only one guest can actually arrive first.
• In problems like this, each guest has an equal chance of arriving first provided no extra information biases the likelihood of their arrival.

Understanding these concepts is crucial for accurately determining probabilities for specific groups of guests, such as uncles, brothers, or cousins.

Sample Space

The **sample space** in probability is the set of all possible outcomes of a trial. In the case of Mrs. Schmulen's gathering, the sample space includes all invited relatives who could be the first to arrive. This comprises the mother, 2 uncles, 3 brothers, and 4 cousins, resulting in a total sample space of 10 distinct members.
Let’s consider the following details:

• The sample space, in this example, consists of 10 possible outcomes since there are 10 guests.
• Each potential outcome has the same probability of occurring if all have an equal chance of arriving first.
• Having a clear view of the sample space is fundamental to computing probabilities of compound events, where you might group certain relatives like the uncles and brothers.

This understanding allows us to define probabilities related to who might be the first guest to arrive at the party efficiently.

Addition Rule

The **addition rule** in probability is essential for calculating the probability of one event or another occurring when the events are mutually exclusive. Simply put, the rule enables us to find the probability that either of two separate events occurs.
Let's apply this to Mrs. Schmulen's party example:

• To find the probability of an uncle or a brother arriving first, add their individual probabilities. This means summing up the individual probabilities because a guest cannot be both an uncle and a brother at once, making these events mutually exclusive.
• Using the formula: If event A has a probability of \( P(A) \) and event B has a probability of \( P(B) \), then the probability of A or B occurring is \( P(A \cup B) = P(A) + P(B) \).
• Therefore, for uncles or brothers: \( P(\text{Uncle or Brother}) = \frac{5}{10} = \frac{1}{2} \).

Utilizing the addition rule simplifies the process of combining probabilities, allowing us to efficiently solve problems involving multiple potential outcomes.