Problem 22 - Basic Statistics for Business and Economics with Student

Question

A study by the National Park Service revealed that 50 percent of vacationers going to the Rocky Mountain region visit Yellowstone Park, 40 percent visit the Tetons, and 35 percent visit both. a. What is the probability a vacationer will visit at least one of these attractions? b. What is the probability .35 called? c. Are the events mutually exclusive? Explain.

Step-by-Step Solution

Answer

a. 0.55; b. Joint probability; c. Not mutually exclusive.

Step 1: Define the Probabilities

First, let's identify the given probabilities in the problem:- Let \( P(Y) \) be the probability of visiting Yellowstone Park, which is 0.50.- Let \( P(T) \) be the probability of visiting the Tetons, which is 0.40.- Let \( P(Y \cap T) \) be the probability of visiting both, which is 0.35.

Step 2: Calculate the Probability of Visiting at Least One

To find the probability that a vacationer will visit at least one of these attractions, we use the formula for the union of two events:\[ P(Y \cup T) = P(Y) + P(T) - P(Y \cap T) \]Substitute the given probabilities into the equation:\[ P(Y \cup T) = 0.50 + 0.40 - 0.35 \]Calculate the result:\[ P(Y \cup T) = 0.55 \]

Step 3: Identify the Probability Name

Probability 0.35 is the probability that a vacationer will visit *both* Yellowstone Park and the Tetons. This is called the **joint probability**.

Step 4: Determine if Events are Mutually Exclusive

Events are mutually exclusive if they cannot occur at the same time. Since there is a probability (0.35) that a vacationer visits both attractions, these events are not mutually exclusive.

Key Concepts

Mutually Exclusive Events

In probability theory, events are classified as mutually exclusive if they cannot occur simultaneously. Imagine you are flipping a coin; getting heads and tails on the same single flip is impossible. Those outcomes are mutually exclusive. In the context of our exercise, we need to determine if visiting Yellowstone Park and the Tetons are mutually exclusive events. That means, can a vacationer visit both places at the same time?

The term mutually exclusive means that the occurrence of one event makes the other impossible. If you have two events, say A and B, they are mutually exclusive if the probability of both A and B occurring together, denoted as \( P(A \cap B) \), is zero. In our exercise, the probability \( P(Y \cap T) = 0.35\), which indicates that 35% of vacationers visit both parks. Since this probability is positive and not zero, the events of visiting Yellowstone and the Tetons are not mutually exclusive. This tells us that a vacationer can indeed visit both attractions on the same trip.

Always remember: mutually exclusive events mean one or the other can occur but not both. In contrast, our example shows that both events can happen together. This is an essential distinction in probability.

Joint Probability

Joint probability refers to the probability of two events occurring simultaneously. It's incredibly useful for determining how two different scenarios can interact. Let's consider the joint probability in the exercise at hand, where 35% of vacationers visit both Yellowstone and the Tetons.

Joint probability is represented by the expression \( P(A \cap B) \), where \( A \) and \( B \) are two events. In this case, our events are visiting Yellowstone Park and visiting the Tetons. We've been given \( P(Y \cap T) = 0.35 \), which means that there is a 35% chance a visitor will see both parks. This joint probability tells us about the overlap between the two events.

Understanding joint probability helps us see the relationship between actions or events. In this context, it means evaluating how likely it is that a vacationer experiences both venues. It's a fundamental concept when examining how different probabilities combine. If you were exploring more complex networks of events, joint probability allows you to map out potential outcomes comprehensively.

Union of Events

The union of events concept in probability involves calculating the probability of either one event or both occurring. It is symbolized as \( A \cup B \) and interpreted as the likelihood that either event A happens, event B happens, or both happen.

To compute the union of events, we use the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] This calculation considers the probability of both events happening twice, so we subtract it to account for that overlap.

In the given problem, vacationers visiting at least one of the attractions – Yellowstone Park or the Tetons – is represented by \( P(Y \cup T) \). Plugging in the numbers, \[ P(Y \cup T) = 0.50 + 0.40 - 0.35 = 0.55 \] indicating a 55% chance that a vacationer visits at least one of the sites.

The union of events thus helps us comprehend how probabilities stack when considering multiple potential outcomes. It elegantly handles overlapping probabilities to give an accurate measure of combined likelihood across different scenarios.