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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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Most popular questions from this chapter

Consider the chance experiment in which both tennis racket head size and grip size are noted for a randomly selected customer at a particular store. The six possible outcomes (simple events) and their probabilities are displayed in the following table: a. The probability that grip size is \(4 \frac{1}{2}\) in. (event \(\mathrm{A}\) ) is $$ P(A)=P\left(O_{2} \text { or } O_{5}\right)=.20+.15=.35 $$ How would you interpret this probability? b. Use the result of Part (a) to calculate the probability that grip size is not \(4 \frac{1}{2}\) in. c. What is the probability that the racket purchased has an oversize head (event \(B\) ), and how would you interpret this probability? d. What is the probability that grip size is at least \(4 \frac{1}{2}\) in.?

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