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Visa Card USA studied how frequently young consumers, ages 18 to \(24,\) use plastic (debit and credit) cards in making purchases (Associated Press, January 16,2006 ). The results of the study provided the following probabilities. \(\bullet\) The probability that a consumer uses a plastic card when making a purchase is .37. \(\bullet\) Given that the consumer uses a plastic card, there is a .19 probability that the consumer is 18 to 24 years old. \(\bullet\) Given that the consumer uses a plastic card, there is a .81 probability that the consumer is more than 24 years old. U.S. Census Bureau data show that \(14 \%\) of the consumer population is 18 to 24 years old. a. Given the consumer is 18 to 24 years old, what is the probability that the consumer uses a plastic card? b. Given the consumer is over 24 years old, what is the probability that the consumer uses a plastic card? c. What is the interpretation of the probabilities shown in parts (a) and (b)? d. Should companies such as Visa, MasterCard, and Discover make plastic cards available to the 18 to 24 years old age group before these consumers have had time to establish a credit history? If no, why? If yes, what restrictions might the companies place on this age group?

Step by step solution

01

Understanding the Probabilities

First, we need to identify the key probabilities provided:1. The probability that a consumer uses a plastic card is \( P(\text{Plastic}) = 0.37 \).2. Given that a consumer uses a plastic card, the probability that the consumer is 18 to 24 years old is \( P(18-24 \mid \text{Plastic}) = 0.19 \).3. Given that a consumer uses a plastic card, the probability that the consumer is more than 24 years old is \( P(>24 \mid \text{Plastic}) = 0.81 \).4. The probability that a consumer is 18 to 24 years old is \( P(18-24) = 0.14 \).

02

Calculate Probability for 18 to 24-Year-Olds Using Bayes' Theorem

We need to find the probability that a consumer uses a plastic card given that they are 18 to 24 years old, \( P(\text{Plastic} \mid 18-24) \). Apply Bayes' Theorem:\[P(\text{Plastic} \mid 18-24) = \frac{P(18-24 \mid \text{Plastic}) \cdot P(\text{Plastic})}{P(18-24)}\]Substitute the known values:\[P(\text{Plastic} \mid 18-24) = \frac{0.19 \times 0.37}{0.14} = 0.5014\]

03

Calculate Probability for Over 24-Year-Olds Using Complement Rule

Note that both younger and older age groups make up the plastic users:\[P(>24 \mid \text{Plastic}) = 0.81\]Use the complement rule, where all probabilities related to \(18-24\) and \(>24\) add up to one under the condition of using a plastic card:\[P(\text{Plastic} \mid >24) = P(\text{Plastic}) = 0.37 \text{ (since not using Bayes', unlike "18-24")}\]However, since part (b) requires an analogous process:Calculate specifically (if needed):\(givendifferencee.g.P(\text{Plastic} \mid >24 )is"notgiven"\):Continuingmore-specificmethod,Bayes'-ishone follows,not basic rule:\[P(\text{Plastic} \mid >24 ) = \frac{P(>24 \mid \text{Plastic}) \cdot P(\text{Plastic})}{1 - P(18-24)}\]Replacing known probabilities:\[P(\text{Plastic} \mid >24) \approx \frac{0.81 \times 0.37}{0.86} \approx 0.348\]

04

Interpretation of Probabilities

The probability \( P(\text{Plastic} \mid 18-24) = 0.5014 \) indicates that about 50.14% of consumers aged 18 to 24 use plastic cards for transactions. Similarly, \( P(\text{Plastic} \mid >24) = 0.348 \) indicates that about 34.8% of consumers older than 24 use plastic cards.

05

Recommendation for Credit Card Companies

Since the probability of young consumers using plastic is higher, companies like Visa might consider targeting this demographic with offerings. However, to manage risk, they could impose restrictions such as setting lower credit limits, offering student cards that require a co-signer, or educational programs about credit use for young adults.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that allows us to update our beliefs about an event based on new evidence. It's particularly useful when dealing with conditional probabilities. In simple terms, it helps you find the probability of an event given that another event has already occurred.

To compute probabilities using Bayes' Theorem, you start by identifying relevant probabilities:

• The prior probability: This is your initial belief or estimate about an event before seeing any new data. In the problem above, the prior probability that a consumer is 18 to 24 years old is given as \( P(18-24) = 0.14 \).
• The likelihood: This is the probability of observing new evidence if your original hypothesis is true. For example, the likelihood that a consumer uses a plastic card given they are between 18 and 24 years old.
• The marginal likelihood: This is the probability of the new evidence under all possible circumstances.

The formula for Bayes' Theorem is:\[P(A \,|\, B) = \frac{P(B \,|\, A) \cdot P(A)}{P(B)}\]Applying this to the problem, we can calculate the probability of using a plastic card when someone is aged 18 to 24, showing how we incorporate new information to adjust our understanding.

Conditional Probability

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. It is a central concept in probability theory and essential for understanding dependencies between events.

To calculate conditional probability, we use the formula:

• \( P(A \,|\, B) = \frac{P(A \cap B)}{P(B)} \)

This formula tells us the probability of event \( A \) happening under the condition that \( B \) has occurred. It means we are focusing on a subset of the sample space determined by \( B \).

In the given exercise, we find two conditional probabilities:

• The probability that a consumer is 18 to 24 years old, given they use a plastic card.
• The probability that a consumer is more than 24 years old, given they use a plastic card.

This idea helps us isolate particular conditions in a dataset and understand relationships, offering insights into customer behaviors.

Data Interpretation

Interpreting data from probability calculations can provide valuable insights into consumer behavior, which is especially useful for companies and marketers. Understanding the probabilities gives us the power to make informed decisions.

The exercise shows how probabilities can reveal patterns, such as:

• A high probability of young consumers between 18 and 24 using plastic cards, indicating their comfort with and reliance on this payment method.
• A lower probability of consumers over 24 using plastic cards, suggesting different behaviors or preferences.

These insights help companies like Visa to consider whether marketing strategies should be adjusted for different age groups. It suggests they might develop products specifically tailored for younger consumers, perhaps with unique features or conditions like limited spending to mitigate risks.

The interpretation of data is a crucial final step in probability exercises, connecting statistical results with practical business strategies.