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Chapter 2: Problem 15

Consider the type of clothes dryer (gas or electric) purchased by each of five different customers at a certain store. a. If the probability that at most one of these purchases an electric dryer is \(.428\), what is the probability that at least two purchase an electric dryer? b. If \(P(\) all five purchase gas \()=.116\) and \(P(\) all five purchase electric) \(=.005\), what is the probability that at least one of each type is purchased?

Short Answer

Expert verified
(a) 0.572; (b) 0.879.

Step by step solution

01

Understanding the Problem Statement

We have probabilities related to customers purchasing gas or electric dryers at a store. We need to find the probability for at least two purchasing electric dryers (part a) and at least one purchasing each type (gas and electric) (part b).
02

Computing Probability for Part (a)

For part (a), we know the probability that at most one purchases an electric dryer is 0.428. This means that the events can include zero or one person purchasing an electric dryer. To find the probability of at least two purchasing an electric dryer, we use the complement rule. The probability of the complement event (at least two) is calculated as the total probability (1) minus the probability of at most one\[ P(\text{at least two electric}) = 1 - P(\text{at most one electric}) = 1 - 0.428 = 0.572. \]
03

Computing Probability for Part (b)

In part (b), the scenario where all purchases are gas or all are electric is given. These two events are complements of the event where there is at least one of each type of dryer.To find the probability that at least one of each type is purchased, we subtract these two mutually exclusive events from the total probability (1):\[ P(\text{at least one of each type}) = 1 - P(\text{all gas}) - P(\text{all electric}) = 1 - 0.116 - 0.005 = 0.879. \]
04

Summarizing Solutions

In part (a), the probability that at least two customers purchase electric dryers is 0.572. In part (b), the probability that at least one of each type of dryer is purchased is 0.879.

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Key Concepts

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

Complement Rule
In probability theory, the complement rule is a simple yet powerful concept used to find the probability of the non-occurrence of an event. This rule states that the probability of an event not happening is equal to one minus the probability of the event occurring. This is represented as \[ P(A') = 1 - P(A) \]where \( P(A') \) is the probability of the complement event, and \( P(A) \) is the probability of the event itself.
Imagine a scenario involving purchasing dryers, as in our exercise. If the probability of purchasing at most one electric dryer is 0.428, then, by the complement rule, the probability that at least two customers are buying electric dryers can be calculated simply as \[ P(\text{at least two electric}) = 1 - 0.428 = 0.572. \] This method is extremely useful when direct calculation seems complex, as it simplifies the process by tackling the complement of the given probability, often making it easier to compute.
Mutually Exclusive Events
Mutually exclusive events are those events in probability that cannot happen simultaneously. When one event happens, it means the other event cannot occur. The probability of the union of mutually exclusive events is simply the sum of their individual probabilities. This can be expressed mathematically as: \[ P(A \cup B) = P(A) + P(B) \] if \( A \) and \( B \) are mutually exclusive.
In our exercise problem, purchasing all gas or all electric dryers are mutually exclusive events. This means customers cannot simultaneously purchase all gas dryers and all electric dryers. Knowing this helps in determining the probability that at least one of each type is bought by using the complement rule, as outlined in the solution, by subtracting the probabilities of the mutually exclusive events from the total probability (1). This technique reliably aids in calculations whenever such distinct, non-overlapping possibilities exist.
Probability Distribution
A probability distribution is a mathematical function that provides the probabilities of the occurrence of different possible outcomes in an experiment. It encompasses the possible values of a random variable and the probability that a random variable falls within each interval of values. Probability distributions are essential frameworks in probability theory, helping to underpin many calculations.
In dealing with dryer purchases, one can assume a probability distribution over purchasing decisions that tells us the likelihood of different combinations of purchases (e.g., all gas, all electric, a mix of both). In part (b) of our problem, the situation assumes probability distributions for these distinct outcomes, like all five customers buying gas or all buying electric dryers. By understanding the distribution of these outcomes, we effectively use the computations to find probabilities of hybrid events such as having at least one gas and one electric dryer, enhancing our grasp of how likely certain combinations are.

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Most popular questions from this chapter

An engineering construction firm is currently working on power plants at three different sites. Let \(A_{i}\) denote the event that the plant at site \(i\) is completed by the contract date. Use the operations of union, intersection, and complementation to describe each of the following events in terms of \(A_{1}, A_{2}\), and \(A_{3}\), draw a Venn diagram, and shade the region corresponding to each one. a. At least one plant is completed by the contract date. b. All plants are completed by the contract date. c. Only the plant at site 1 is completed by the contract date. d. Exactly one plant is completed by the contract date. e. Either the plant at site 1 or both of the other two plants are completed by the contract date.

An academic department with five faculty members narrowed its choice for department head to either candidate \(A\) or candidate \(B\). Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for \(A\) and two for \(B\). If the slips are selected for tallying in random order, what is the probability that \(A\) remains ahead of \(B\) throughout the vote count (e.g., this event occurs if the selected ordering is \(A A B A B\), but not for \(A B B A A)\) ?

A certain system can experience three different types of defects. Let \(A_{i}(i=1,2,3)\) denote the event that the system has a defect of type \(i\). Suppose that $$ \begin{aligned} &P\left(A_{1}\right)=.12 \quad P\left(A_{2}\right)=.07 \quad P\left(A_{3}\right)=.05 \\ &P\left(A_{1} \cup A_{2}\right)=.13 \quad P\left(A_{1} \cup A_{3}\right)=.14 \\\ &P\left(A_{2} \cup A_{3}\right)=.10 \quad P\left(A_{1} \cap A_{2} \cap A_{3}\right)=.01 \end{aligned} $$ a. What is the probability that the system does not have a type 1 defect? b. What is the probability that the system has both type 1 and type 2 defects? c. What is the probability that the system has both type 1 and type 2 defects but not a type 3 defect? d. What is the probability that the system has at most two of these defects?

An electronics store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component: Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood Compact disc player: Onkyo, Pioneer, Sony, Technics Speakers: Boston, Infinity, Polk Turntable: Onkyo, Sony, Teac, Technics A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions: a. In how many ways can one component of each type be selected? b. In how many ways can components be selected if both the receiver and the compact disc player are to be Sony? c. In how many ways can components be selected if none is to be Sony? d. In how many ways can a selection be made if at least one Sony component is to be included? e. If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component?

A sonnet is a 14-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just 10 sonnets, each on a different page. However, these were structured such that other sonnets could be created as follows: the first line of a sonnet could come from the first line on any of the 10 pages, the second line could come from the second line on any of the 10 pages, and so on (successive lines were perforated for this purpose). a. How many sonnets can be created from the 10 in the book? b. If one of the sonnets counted in part (a) is selected at random, what is the probability that none of its lines came from either the first or the last sonnet in the book?
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