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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

Consider a system consisting of four components, as pictured in the following diagram: Components 1 and 2 form a series subsystem, as do Components 3 and 4 . The two subsystems are connected in parallel. Suppose that \(P(1\) works \()=.9, P(2\) works \()=.9\), \(P(3\) works \()=.9\), and \(P(4\) works \()=.9\) and that the four components work independently of one another. a. The \(1-2\) subsystem works only if both components work. What is the probability of this happening? b. What is the probability that the \(1-2\) subsystem doesn't work? that the \(3-4\) subsystem doesn't work? c. The system won't work if the \(1-2\) subsystem doesn't work and if the \(3-4\) subsystem also doesn't work. What is the probability that the system won't work? that it will work? d. How would the probability of the system working change if a \(5-6\) subsystem were added in parallel with the other two subsystems? e. How would the probability that the system works change if there were three components in series in each of the two subsystems?

After all students have left the classroom, a statistics professor notices that four copies of the text were left under desks. At the beginning of the next lecture, the professor distributes the four books at random to the four stu- dents \((1,2,3\), and 4\()\) who claim to have left books. One possible outcome is that 1 receives 2 's book, 2 receives 4 's book, 3 receives his or her own book, and 4 receives 1 's book. This outcome can be abbreviated \((2,4,3,1)\). a. List the 23 other possible outcomes. b. Which outcomes are contained in the event that exactly two of the books are returned to their correct owners? As- suming equally likely outcomes, what is the probability of this event? c. What is the probability that exactly one of the four students receives his or her own book? d. What is the probability that exactly three receive their own books? e. What is the probability that at least two of the four students receive their own books?

A certain university has 10 vehicles available for use by faculty and staff. Six of these are vans and four are cars. On a particular day, only two requests for vehicles have been made. Suppose that the two vehicles to be assigned are chosen in a completely random fashion from among the 10 . a. Let \(E\) denote the event that the first vehicle assigned is a van. What is \(P(E)\) ? b. Let \(F\) denote the probability that the second vehicle assigned is a van. What is \(P(F \mid E)\) ? c. Use the results of Parts (a) and (b) to calculate \(P(E\) and \(F)\) (Hint: Use the definition of \(P(F \mid E) .)\)

6.3 Consider the chance experiment in which the type of transmission- automatic (A) or manual (M) - is recorded for each of the next two cars purchased from a certain dealer. a. What is the set of all possible outcomes (the sample space)? b. Display the possible outcomes in a tree diagram. c. List the outcomes in each of the following events. Which of these events are simple events? i. \(B\) the event that at least one car has an automatic transmission ii. \(C\) the event that exactly one car has an automatic transmission iii. \(D\) the event that neither car has an automatic transmission d. What outcomes are in the event \(B\) and \(C\) ? In the event \(B\) or \(C\) ?

At a large university, the Statistics Department has tried a different text during each of the last three quarters. During the fall quarter, 500 students used a book by Professor Mean; during the winter quarter, 300 students used a book by Professor Median; and during the spring quarter, 200 students used a book by Professor Mode. A survey at the end of each quarter showed that 200 students were satisfied with the text in the fall quarter, 150 in the winter quarter, and 160 in the spring quarter. a. If a student who took statistics during one of these three quarters is selected at random, what is the probability that the student was satisfied with the textbook? b. If a randomly selected student reports being satisfied with the book, is the student most likely to have used the book by Mean, Median, or Mode? Who is the least likely author? (Hint: Use Bayes' rule to compute three probabilities.)
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