Problem 21 - Basic Statistics for Business and Economics with Student

Question

A survey of grocery stores in the Southeast revealed 40 percent had a pharmacy, 50 percent had a floral shop, and 70 percent had a deli. Suppose 10 percent of the stores have all three departments, 30 percent have both a pharmacy and a deli, 25 percent have both \(a\) floral shop and deli, and 20 percent have both a pharmacy and floral shop. a. What is the probability of selecting a store at random and finding it has both a pharmacy and a floral shop? b. What is the probability of selecting a store at random and finding it has both a pharmacy and a deli? c. Are the events "select a store with a deli" and "select a store with a pharmacy" mutually exclusive? d. What is the name given to the event of "selecting a store with a pharmacy, a floral shop, and a deli?" e. What is the probability of selecting a store that does not have all three departments?

Step-by-Step Solution

Answer

a. 0.20 b. 0.30 c. No d. Intersection of events e. 0.90

Step 1: Identifying Known Probabilities

The given probabilities are: - Probability of a store having a pharmacy, \( P(P) = 0.40 \).- Probability of a store having a floral shop, \( P(F) = 0.50 \).- Probability of a store having a deli, \( P(D) = 0.70 \).- Probability of a store having all three departments, \( P(P \cap F \cap D) = 0.10 \).- Probability of a store having both a pharmacy and a deli, \( P(P \cap D) = 0.30 \).- Probability of a store having both a floral shop and a deli, \( P(F \cap D) = 0.25 \).- Probability of a store having both a pharmacy and a floral shop, \( P(P \cap F) = 0.20 \).

Step 2: Calculating Probability of Pharmacy and Floral Shop

For part (a), we need the probability of a store having both a pharmacy and a floral shop, which is already given as \( P(P \cap F) = 0.20 \).

Step 3: Probability of Pharmacy and Deli

For part (b), the probability of a store having both a pharmacy and a deli is already provided as \( P(P \cap D) = 0.30 \).

Step 4: Checking for Mutual Exclusivity

For part (c), events are mutually exclusive if they cannot happen at the same time. However, since stores can have both a deli and a pharmacy (as given by the conditional probabilities above), these events are **not** mutually exclusive.

Step 5: Naming the Event of All Three

For part (d), the event of selecting a store with a pharmacy, a floral shop, and a deli is commonly called the intersection of all three events, or having all three departments, noted as \( P(P \cap F \cap D) \).

Step 6: Calculating Probability of Not Having All Three Departments

For part (e), we use the complement rule. The probability of not having all three departments is \( 1 - P(P \cap F \cap D) = 1 - 0.10 = 0.90 \).

Key Concepts

Mutually Exclusive Events

When learning about probability, it's important to understand the concept of mutually exclusive events. These are events that cannot occur at the same time. Think of tossing a coin: you cannot land on both heads and tails simultaneously.
This is what makes them mutually exclusive.
In our grocery store exercise, we need to determine if having a pharmacy and a deli in the same store are mutually exclusive.
Since it's possible for a store to have both departments, as shown by the given probabilities, we identify that these events are not mutually exclusive. This distinction is crucial for understanding how different probabilities relate to one another in scenarios where multiple conditions can overlap.

Intersection of Events

The intersection of events is a key concept in understanding how multiple events can occur together. In probability, the intersection is represented by the symbol \( \cap \), which refers to the 'and' condition. When calculating the intersection of two events, we're interested in the probability that both events happen at the same time.

In the example of the grocery stores, several probabilities concerning intersections are given. For instance:

• The probability of a store having both a pharmacy and a floral shop is \( P(P \cap F) = 0.20 \).
• Another intersection, having both a pharmacy and a deli, is \( P(P \cap D) = 0.30 \).

These intersecting probabilities reveal how frequently these combinations occur, and are useful for more complex probability calculations.

Complement Rule

The complement rule in probability is a handy tool when you want to find the chance of an event not happening. Mathematically, the complement of an event \( A \) is denoted as \( A^c \) and is calculated as \( 1 - P(A) \). This rule is especially useful when it’s easier to calculate what you don’t want, rather than what you do want.

In our exercise, we applied the complement rule for a store not having all three departments (pharmacy, floral shop, and deli). Given that the probability of having all three is \( P(P \cap F \cap D) = 0.10 \), the complement rule helps us find the probability of not having all three:

• \( 1 - P(P \cap F \cap D) = 0.90 \)

This calculation is simple, yet powerful for deriving probabilities in various situations.

Conditional Probability

Although conditional probability isn't directly used in solving the grocery store problem, it's a powerful concept you should understand. Conditional probability measures the probability of an event occurring, given that another event has already occurred.

The notation used is \( P(A \mid B) \), where you are calculating the probability of \( A \) happening, assuming \( B \) is true. Understanding conditional probabilities is crucial for dealing with dependent events in probability theory.
While our exercise did not need this calculation directly, being aware of conditional probabilities allows you to think critically about how one event impacts another. As you delve deeper into probability, you'll see more complex problems where this concept is essential.