Problem 25 - A Graphical Approach to College Algebra

Question

The following table shows the probability that a customer at a department store will make a purchase in the indicated price range.$$\begin{array}{|l|c|} \hline \multicolumn{1}{|c|}\text { cost } & \text { Probability } \\ \hline \text { Below \$5 } & 0.25 \\ \$ 5-\$ 19.99 & 0.37 \\ \$ 20-\$ 39.99 & 0.11 \\ \$ 40-\$ 69.99 & 0.09 \\ \$ 70-\$ 99.99 & 0.07 \\ \$ 100-\$ 149.99 & 0.08 \\ \$ 150 \text { or more } & 0.03 \end{array}$$ Find the probability that a customer makes a purchase that is (a) less than \(\$ 20\) (b) \(\$ 40\) or more. (c) more than \(\$ 99.99\) (d) less than \(\$ 100\).

Step-by-Step Solution

Answer

(a) 0.62, (b) 0.27, (c) 0.11, (d) 0.89.

Step 1: Understanding Given Data

First, we need to understand the table which provides the probability for purchases in different price ranges. This information will be used to calculate probabilities for the specified conditions.

Step 2: Calculating Probability Less Than $20 (a)

We need to find the probability of making a purchase of less than \(20. According to the table, this includes two price ranges: 'Below \)5' and '\(5-\)19.99'. We sum their probabilities: \(0.25 + 0.37 = 0.62\).

Step 3: Calculating Probability of $40 or More (b)

For purchases \(40 or more, the price ranges included are: '\)40-\(69.99', '\)70-\(99.99', '\)100-\(149.99', and '\)150 or more'. We sum their probabilities: \(0.09 + 0.07 + 0.08 + 0.03 = 0.27\).

Step 4: Calculating Probability More Than $99.99 (c)

For purchases more than \(99.99, the applicable ranges are: '\)100-\(149.99' and '\)150 or more'. We sum these probabilities: \(0.08 + 0.03 = 0.11\).

Step 5: Calculating Probability Less Than $100 (d)

For purchases less than \(100, we consider price ranges: 'Below \)5', '\(5-\)19.99', '\(20-\)39.99', '\(40-\)69.99', and '\(70-\)99.99'. We sum these probabilities: \(0.25 + 0.37 + 0.11 + 0.09 + 0.07 = 0.89\).

Key Concepts

Conditional Probability

Conditional probability is a measure of the probability of an event occurring, given that another event has already occurred. For instance, if you know the probability of buying a product below $20 is 0.62, you can use conditional probability to refine your prediction about a future purchase based on new information. This concept helps in narrowing down possibilities when additional conditions or criteria are met.
To calculate conditional probability, you use the formula: \[P(A|B) = \frac{P(A \cap B)}{P(B)}\]where \(P(A|B)\) is the probability of event A given event B, \(P(A \cap B)\) is the probability of both events occurring together, and \(P(B)\) is the probability of event B. This formula captures how information about event B might affect observations of event A.
Understanding conditional probability is especially useful in fields like quality control and medical testing, where you want to know the probability of an event happening (like a machine producing a faulty item or a patient having a disease), given that certain conditions have been met. Thus, it vastly enhances decision-making under uncertainty.

Probability Distribution

A probability distribution describes how probabilities are distributed over the values of a random variable. In the context of the department store problem, each purchase price range represents a value that the random variable can take, and the associated probabilities indicate how likely each purchase range is. This helps visualize how likely different outcomes are.
There are different types of probability distributions, such as discrete and continuous distributions. Discrete distributions deal with outcomes that can be counted, like our purchase ranges. Continuous distributions, on the other hand, involve outcomes that can take any value within a range, such as heights or weights. Our table of purchase probabilities is a discrete probability distribution because each outcome (purchase range) has a specific probability assigned to it.
Building a probability distribution requires ensuring all probabilities sum up to 1. In the given exercise, the purchase probabilities total to 1 (i.e., \(0.25 + 0.37 + 0.11 + 0.09 + 0.07 + 0.08 + 0.03 = 1\)). This confirmation is crucial for a valid probability distribution and allows you to make accurate predictions about events.

Statistics in Mathematics

Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data. It is essential in making informed decisions based on data patterns and trends. In the given exercise, statistics help us understand customer purchasing behavior at a department store through probability calculations and distributions.
Statistics often involve measures like averages and percentiles to summarize data, which can then be analyzed for trends or variances. Basic statistical operations also include calculating probabilities, like determining how many customers fall into each purchasing category. This process involves both descriptive and inferential statistics. Descriptive statistics summarize raw data, like the probability table that shows potential spending habits. Inferential statistics allow us to make predictions or inferences about a population based on a dataset.
For example, using statistical analyses, a store manager could extrapolate from this table to predict customer purchasing patterns, helping with inventory decisions. Thus, statistics forms a backbone for research, business strategies, and even everyday decisions, using data to draw sensible conclusions.