Problem 33 - Algebra for College Students

Question

The probability that a customer in a department store will buy a blouse is \(0.15\), the probability that she will buy a pair of shoes is \(0.10\), and the probability that she will buy both a blouse and a pair of shoes is \(0.05\). Find the probability that the customer will buy a blouse, given that she has already purchased a pair of shoes. Also find the probability that she will buy a pair of shoes, given that she has already purchased a blouse.

Step-by-Step Solution

Answer

The probability of buying a blouse given shoes is 0.5, and of buying shoes given a blouse is approximately 0.333.

Step 1: Identify Known Probabilities

We are given three probabilities: \( P(B) = 0.15 \) for buying a blouse, \( P(S) = 0.10 \) for buying shoes, and \( P(B \cap S) = 0.05 \), which is the probability of buying both a blouse and shoes.

Step 2: Conditional Probability Formula

We need to find two conditional probabilities: \( P(B|S) \) and \( P(S|B) \). The formula for conditional probability is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \).

Step 3: Calculate Probability of Buying a Blouse Given Shoes

Using the formula \( P(B|S) = \frac{P(B \cap S)}{P(S)} \), substitute the known values: \( P(B|S) = \frac{0.05}{0.10} = 0.5 \).

Step 4: Calculate Probability of Buying Shoes Given a Blouse

Similarly, use the formula \( P(S|B) = \frac{P(B \cap S)}{P(B)} \), substitute the known values: \( P(S|B) = \frac{0.05}{0.15} = \frac{1}{3} \approx 0.333 \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random events. It is used to quantify uncertainty and model real-world situations where chance plays a crucial role. In simple terms, probability measures how likely an event is to occur.
To help understand probabilities, consider these basic concepts:

• Experiment: An action or process that produces unpredictable outcomes. For example, buying items at a store.
• Sample Space: The set of all possible outcomes of an experiment. In our example, this could involve purchasing different combinations of items like clothes or shoes.
• Event: A specific outcome or combination of outcomes. Events are what we calculate probabilities for, like buying a blouse.

We often assign probabilities between 0 and 1 to events.
Events with a probability of 0 never occur, while events with a probability of 1 always occur.

Joint Probability

Joint probability refers to the likelihood of two events happening at the same time. It's integral to understanding more complex probability scenarios, especially in real-world applications.
Imagine visiting a department store and wanting to know the chance of buying both a blouse and a pair of shoes in the same trip.
This would involve calculating their joint probability, denoted as:\[ P(B \cap S) \]Where:

• \( B \): Event of purchasing a blouse.
• \( S \): Event of purchasing shoes.
• \( P(B \cap S) = 0.05 \): The joint probability that both a blouse and shoes are purchased.

The calculation of joint probability is especially useful when trying to determine the relationship between two dependent events.
It serves as the foundation for calculating conditional probabilities.

Blouse and Shoes Purchase Probability

Understanding the specific likelihood of purchasing a blouse and shoes involves concepts of conditional probability.
Conditional probability is the probability of an event occurring given that another event has already occurred.
In this scenario, we focus on two main calculations:

• Probability of buying a blouse given the purchase of shoes: Using the formula \[ P(B|S) = \frac{P(B \cap S)}{P(S)} \]Plug in the respective probabilities to find:\[ P(B|S) = \frac{0.05}{0.10} = 0.5 \]
• Probability of buying shoes given the purchase of a blouse: The formula here is:\[ P(S|B) = \frac{P(B \cap S)}{P(B)} \]Substituting the given values gives:\[ P(S|B) = \frac{0.05}{0.15} = 0.333 \]

These calculations help evaluate how one purchase can influence the likelihood of making another purchase.
It's important in decision-making and for understanding consumer behavior in a shopping context.