91Ó°ÊÓ

The manager of a music store has kept records of the number of CDs bought in a single transaction by customers who make a purchase at the store. The accompanying table gives six possible outcomes and the estimated probability associated with each of these outcomes for the chance experiment that consists of observing the number of CDs purchased by the next customer at the store. $$ \begin{aligned} &\begin{array}{l} \text { Number of CDs } \\ \text { purchased } \end{array} & 1 & 2 & 3 & 4 & 5 & 6 \text { or more } \\ &\begin{array}{c} \text { Estimated } \\ \text { probability } \end{array} & .45 & .25 & .10 & .10 & .07 & .03 \end{aligned} $$ a. What is the estimated probability that the next customer purchases three or fewer CDs? b. What is the estimated probability that the next customer purchases at most three CDs? How does this compare to the probability computed in Part (a)? c. What is the estimated probability that the next customer purchases five or more CDs? d. What is the estimated probability that the next customer purchases one or two CDs? e. What is the estimated probability that the next customer purchases more than two CDs? Show two different ways to compute this probability that use the probability rules of this section.

Step by step solution

01

Calculate probability for three or fewer CDs

Three or fewer CDs means 1, 2 or 3 CDs. So add up the probability of buying 1, 2 or 3 CDs. \( .45 + .25 + .10 = .80 \)

02

Compare the above result with a. and b.

The estimated probability that a customer purchases at most three CDs is the same as the estimated probability that a customer purchases three or fewer CDs. In both cases, we add up the probabilities of the customer purchasing 1, 2 or 3 CDs, which equals to .80.

03

Calculate probability for five or more CDs

Five or more CDs means 5 CDs, 6 CDs or more. So add up the probability of buying 5 CDs or 6 CDs or more. \( .07 + .03 = .10 \)

04

Calculate probability for one or two CDs

One or two CDs means 1 or 2 CDs. So add up the probability of buying 1 or 2 CDs. \( .45 + .25 = .70 \)

05

Calculate probability for more than two CDs

More than two CDs means buying 3, 4, 5 or 6 CDs or more. So add up these probability. \( .10 + .10 + .07 + .03 = .30 \). Another way to get this is to subtract the sum of probabilities for 1 or 2 CDs from total probability of 1. \( 1 - .70 = .30 \)

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Key Concepts

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

Discrete Probability Distribution

In probability theory, a discrete probability distribution is a type of statistical distribution which shows the probabilities of outcomes with finite values. For example, in our CD store scenario, the possible number of CDs purchased - ranging from 1 to 6 or more - represents discrete numerical variables.
Each of these outcomes has an associated probability, reflecting how likely it is for each outcome to occur during a single trial or observation.
In this case, the probabilities provided in the exercise, such as 0.45 for purchasing 1 CD, form the probability distribution of the experiment.
This model allows us to visualize and calculate probabilities for specific outcomes within a finite, countable range.

• All probabilities in a discrete distribution must sum to 1, acknowledging that one of the outcomes must occur.
• Each probability is a number between 0 and 1, where 0 indicates an impossible outcome and 1 gives certainty.
• These sums allow us to answer questions like, "What is the probability of 3 or fewer CDs being purchased?" by summing relevant probabilities (e.g., 0.45 + 0.25 + 0.10).

Probability Rules

Probability rules are essential guidelines that assist us in computing and understanding different probabilistic scenarios.
One basic rule is that the sum of the probabilities of all possible distinct outcomes must equal 1.
By leveraging these rules, we can figure out not only the chance of singular outcomes but also more complex combinations.
Using the complement rule, when calculating the probability of an event, you'd subtract the probability of its complement from 1.

• For instance, to determine the probability of a customer purchasing more than two CDs, we calculate the complement of the event that a customer buys 1 or 2 CDs (i.e., 1 - 0.70 = 0.30).
• The addition rule helps to find the probability of events that can be mutually exclusive, like purchasing 1 or 2 CDs (computed by adding their probabilities, 0.45 + 0.25 = 0.70).
• Such rules are instrumental for creating clear approaches to solving probability-related questions, as seen in our CD example.

Calculating Cumulative Probability

Cumulative probability computes the likelihood of a variable falling within or below a certain range or threshold.
It's particularly useful in assessing the probability of multiple outcomes.
In the CD purchase scenario, let's see how cumulative probability functions in practice.

• To find the probability that a customer purchases "three or fewer" CDs, you'd add the individual probabilities for purchasing 1, 2, or 3 CDs: 0.45 + 0.25 + 0.10 = 0.80.
• Simultaneously, "at most three" CDs has the same probability because both calculations encompass the possibilities of 1 to 3 CDs.
• Cumulative probability tells us about the sum of all probabilities up to the desired event, helping in situations where you're interested in a range of outcomes rather than a specific single outcome.

These types of calculations form the backbone of understanding cumulative distribution functions and aid in clearer interpretation of data.