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Chapter 6: Problem 8

Suppose that, starting at a certain time, batteries coming off an assembly line are examined one by one to see whether they are defective (let \(\mathrm{D}=\) defective and \(\mathrm{N}=\) not defective). The chance experiment terminates as soon as a nondefective battery is obtained. a. Give five possible experimental outcomes. b. What can be said about the number of outcomes in the sample space? c. What outcomes are in the event \(E\), that the number of batteries examined is an even number?

Short Answer

Expert verified
The five given experimental outcomes were: [(N)], [(D, N)], [(D, D, N)], [(D, D, D, N)], [(D, D, D, D, N)]. The number of outcomes in the sample space is infinite, as the sequence could be of any length as long as it ends with a 'N'. Outcomes in event E are those where the number of batteries examined is even; therefore, any sequence with an odd number of D's before the terminating N is in E.

Step by step solution

01

Define an outcome

An outcome for this experiment is a sequence from start until a non-defective battery is found. Thus, an outcome could include several defective batteries (D) followed by a single non-defective battery (N). Note, however, that the sequence must always end with a non-defective battery.
02

Give five possible outcomes

The sequence can be of varying lengths, but always ends with a 'N'. Here are five possible outcomes: \[[(N)], [(D, N)], [(D, D, N)], [(D, D, D, N)], [(D, D, D, D, N)]\]. These represent five cases: finding a non-defective battery on the first try, second try, third try, fourth try, and fifth try respectively.
03

Consider the number of outcomes in the sample space

As the sequence can be of any length as long as it ends with a 'N', the size of the sample space is infinite.
04

Identify the outcomes constituting event E

The event E states that the number of batteries examined is an even number. That means, the length of the sequence must be even. Looking at the sample outcomes, we see these are included in E: \[[(D, N)], [(D, D, D, N)], ...\]. In general any sequence with an odd number of D's before the terminating N is in E.

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Key Concepts

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

Sample Space
In probability, the sample space is the set of all possible outcomes of a probability experiment. For our battery example, an outcome is a sequence of batteries examined until a non-defective one is found. Each sequence ends with an 'N', indicating a non-defective battery.

The beauty of sample spaces is they provide a comprehensive view of every outcome that can occur in an experiment. In this scenario, the sequence length can be infinite because it only stops when you get a non-defective battery. So, you can have
  • (N) - A non-defective found right away.
  • (D, N) - A defective battery followed by a non-defective one.
  • (D, D, N) - Two defectives followed by a non-defective.
As you can see, the list goes on without end, making our sample space infinitely large!
Experimental Outcomes
An experimental outcome is the result of a single trial within the sample space. In the battery example, outcomes are sequences that always terminate with a non-defective battery. This is critical because we only stop our experiment when we have at least one 'N'.

Consider these example outcomes:
  • (N) - If the first battery is non-defective, this is the outcome.
  • (D, N) - The first is defective, but the second isn't.
  • (D, D, D, N) - Three defective ones before landing a good one.
Outcomes are crucial as they define the sequences possible in an experiment, showing us the various ways events can unfold.
Event Probability
Event probability refers to the likelihood of a specific event occurring within the sample space. An event is a set of outcomes that share a common property. For this task, our event of interest is the number of batteries examined being even.

To identify this, count the number of batteries checked in each sequence. Our event occurs when this number is even. Consider these cases:
  • (D, N) - Two batteries examined, so this event occurs.
  • (D, D, D, N) - Four batteries examined, another occurrence of our event.
In formal terms, any sequence characterized by an odd number of 'D's (defectives) followed by one 'N' fits our event requirement. Understanding event probability helps predict the chances of different outcomes, aiding in decision-making and planning.

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Most popular questions from this chapter

According to a study conducted by a risk assessment firm (Associated Press, December 8,2005 ), drivers residing within one mile of a restaurant are \(30 \%\) more likely to be in an accident in a given policy year. Consider the following two events: \(A=\) event that a driver has an accident during a policy year \(R=\) event that a driver lives within one mile of a restaurant Which of the following four probability statements is consistent with the findings of this survey? Justify your choice. i. \(P(A \mid R)=.3\) iii. \(\frac{P(A \mid R)}{P\left(A \mid R^{C}\right)}=.3\) ii. \(P\left(A \mid R^{C}\right)=.3 \quad\) iv. \(\frac{P(A \mid R)-P\left(A \mid R^{C}\right)}{P\left(A \mid R^{C}\right)}=.3\)

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A shipment of 5000 printed circuit boards contains 40 that are defective. Two boards will be chosen at random, without replacement. Consider the two events \(E_{1}=\) event that the first board selected is defective and \(E_{2}=\) event that the second board selected is defective. a. Are \(E_{1}\) and \(E_{2}\) dependent events? Explain in words. b. Let not \(E_{1}\) be the event that the first board selected is not defective (the event \(E_{1}^{C}\) ). What is \(P\left(\right.\) not \(\left.E_{1}\right)\) ? c. How do the two probabilities \(P\left(E_{2} \mid E_{1}\right)\) and \(P\left(E_{2} \mid\right.\) not \(\left.E_{1}\right)\) compare? d. Based on your answer to Part (c), would it be reasonable to view \(E_{1}\) and \(E_{2}\) as approximately independent?

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