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Chapter 5: Problem 83

Fill 鈥檈r up! In a recent month, 88% of automobile drivers filled their vehicles with regular gasoline, 2% purchased midgrade gas, and 10% bought premium gas.17 Of those who bought regular gas, 28% paid with a credit card; of customers who bought midgrade and premium gas, 34% and 42%, respectively, paid with a credit card. Suppose we select a customer at random. Draw a tree diagram to represent this situation. What鈥檚 the probability that the customer paid with a credit card? Use the four-step process to guide your work.

Short Answer

Expert verified
Probability that a customer paid with a credit card is 29.52%.

Step by step solution

01

Understand the Problem

We need to find the probability that a randomly selected customer paid with a credit card, given probabilities for different types of gasoline purchases and the likelihood of using a credit card for each type. The types of gasoline are regular, midgrade, and premium.
02

Draw a Tree Diagram

Create a tree diagram to visually represent the situation. Start with a node for gasoline type, splitting into branches for regular (88%), midgrade (2%), and premium (10%). Each branch then splits into whether the customer paid with a credit card or not, with the associated probabilities: Regular (28% credit card, 72% not), Midgrade (34% credit card, 66% not), and Premium (42% credit card, 58% not).
03

Calculate the Probabilities for Each Branch

For each branch that includes paying with a credit card, calculate the joint probability by multiplying the probability of choosing that gasoline type by the probability of paying with a credit card. For regular gasoline: 0.88 * 0.28 = 0.2464. For midgrade gasoline: 0.02 * 0.34 = 0.0068. For premium gasoline: 0.10 * 0.42 = 0.042.
04

Sum the Probabilities

Add up the probabilities calculated in Step 3 to find the total probability of a customer paying with a credit card. Add the three probabilities: 0.2464 + 0.0068 + 0.042 = 0.2952.

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Key Concepts

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

Tree Diagram
A tree diagram is a visual tool that helps organize and represent probabilities in a structured way.
They are especially useful in situations where multiple stages or categorical choices impact the outcome. In our scenario regarding gasoline purchases, the tree diagram illustrates the sequence of customer decisions from choosing a type of gasoline to deciding the method of payment.
Let's break this down:
  • Start with a single initial node that branches out into nodes for each choice of gasoline: regular, midgrade, and premium.
  • Each of these nodes then expands into further branches representing payment choices: paying with a credit card or not.
  • The probability for each path is shown along the branch, and these paths extend from the start to the end of the tree.
By structuring the problem with a tree diagram, it becomes easier to calculate joint probabilities for specific outcomes and visualize the full range of potential customer choices.
Credit Card Payment Probability
Probabilities tell us the likelihood of a certain event occurring.
In this exercise, we want to determine the total probability that a customer paid with a credit card at the gas pump, depending on the type of gasoline they purchased.
Here's how to find it:
  • Multiply the probability of selecting each type of gasoline by the probability of paying with a credit card for that specific type.
  • For regular gasoline: calculate 0.88 (probability of regular) multiplied by 0.28 (paying with card) = 0.2464.
  • For midgrade gasoline: calculate 0.02 multiplied by 0.34 = 0.0068.
  • For premium gasoline: calculate 0.10 multiplied by 0.42 = 0.0420.
Finally, sum these joint probabilities: 0.2464 + 0.0068 + 0.042 to get the total probability of credit card payment, which is 0.2952.
This means there's about a 29.52% chance that a randomly chosen customer paid with a credit card.
Gasoline Purchases
In our exercise, customers can choose from three gasoline types: regular, midgrade, or premium.
Understanding the probability of selecting each type of gasoline is important for calculating the overall probability of subsequent events, like using a credit card for payment.
Consider these details:
  • Regular gasoline is the most common, with a selection probability of 88% (0.88). This very high percentage indicates that most drivers prefer regular gasoline for its cost-effectiveness and compatibility with many cars.
  • Midgrade gasoline sees much less frequent purchases at 2% (0.02), suggesting it's less popular among consumers compared to other options.
  • Premium gasoline is chosen by 10% (0.10) of customers, often preferred by those with high-performance vehicles or when recommended for specific engines.
These probabilities form the foundational base of our tree diagram and affect all subsequent calculations related to payment choices.

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