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Chapter 5: Problem 16

Your teacher gives a truefalse pop quiz with 10 questions. a. Show that the number of possible outcomes for the sample space of possible sequences of 10 answers is 1024 . b. What is the complement of the event of getting at least one of the questions wrong? c. With random guessing, show that the probability of getting at least one question wrong is \(0.999 .\)

Short Answer

Expert verified
There are 1,024 possible sequences; the complement of getting at least one wrong is getting all answers right; the probability of getting at least one wrong is 0.999.

Step by step solution

01

Determining Possible Outcomes

In a true or false quiz, each question has two possible answers: true or false. Therefore, for 10 questions, the total number of possible combinations of answers is calculated using the formula for permutations with repetition, which is given by \[ 2^{10} = 1024 \]Thus, there are 1024 possible sequences of answers.
02

Understanding the Complement Event

The event "getting at least one question wrong" refers to sequences where one or more answers are incorrect. The complement of this event is the sequence where all answers are correct. This means the student must get all 10 answers correct.
03

Calculating Complement Probability

When guessing randomly, each question has a probability of \( \frac{1}{2} \) of being correct. Thus, the probability of getting all answers correct is \[ \left( \frac{1}{2} \right)^{10} = \frac{1}{1024} \]
04

Finding Complement's Probability Value

The probability of getting at least one question wrong is the complement of the probability of getting all answers correct. Therefore,\[ P(\text{at least one wrong}) = 1 - P(\text{all correct}) = 1 - \frac{1}{1024} = \frac{1023}{1024} \approx 0.999 \]

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Key Concepts

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

Permutation with Repetition
When it comes to arranging objects, permutations are all about order. For a true-false quiz with 10 questions, each question can be either "true" or "false"鈥攖hat鈥檚 two possibilities per question. Since each of the 10 questions allows for either response, this is a classic scenario of "permutation with repetition."

  • In permutations with repetition, the formula to calculate the total number of outcomes is given by \( n^r \). Here, \( n \) is the number of choices for each event, and \( r \) is the number of events.
  • In our quiz, \( n = 2 \) (for true or false) and \( r = 10 \) (questions), thus \( 2^{10} = 1024 \) outcomes.
This means there are 1024 possible sequences of answers for this set of questions. Each sequence is a unique combination of correct and incorrect responses.
Complement Event
Probability theory often involves discussing outcomes and their opposites. The complement of an event includes all the scenarios that are not part of our event of interest. For instance, if we seek the probability of something not happening, we explore its complement.

  • The event "getting at least one question wrong" implies having sequences where some answers are incorrect.
  • The complement is "getting all questions right," meaning every answer is correct.
Understanding complements is crucial as it helps in calculating probabilities indirectly. By finding the probability of the complement, we can easily determine the probability of our desired event by subtracting from 1.
Sample Space
The idea of a sample space is essential in probability theory. It's the set of all possible outcomes. For this quiz, our sample space consists of all the possible sequences of true and false answers across the 10 questions.

  • This sample space includes all permutations of answers, considering each question independently as true or false.
  • Given 10 questions, each with 2 possible answers, there are \( 2^{10} = 1024 \) possible outcomes.
Each combination in the sample space represents a specific sequence, an essential concept allowing us to calculate probabilities for various scenarios, such as the likelihood of getting all questions correct or at least one wrong.
Random Guessing
Randomly guessing an answer is like flipping a coin鈥攊t鈥檚 a 50/50 chance. For each question in a true-false quiz, there is a \( \frac{1}{2} \) probability of guessing correctly.

  • Random guessing means each question remains independent, with a fresh \( \frac{1}{2} \) chance for both "true" and "false."
  • To get all questions right by guessing, the probability is \( \left( \frac{1}{2} \right)^{10} = \frac{1}{1024} \).
This shows the strength in using probability theory to understand outcomes. While the chance of guessing all answers correctly is slim, the likelihood of missing at least one is much higher, calculated as the complement: \( \frac{1023}{1024} \) or approximately 0.999.

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Most popular questions from this chapter

Workers specified as actively disengaged are those who are emotionally disconnected from their work and workplace. A Gallup poll conducted in December \(2010^{5}\) surveyed individuals who were either unemployed or who were actively disengaged in their current position. Individuals were asked to classify themselves as thriving or struggling. The poll reported that \(42 \%\) of the actively disengaged group claimed to be thriving, compared to \(48 \%\) of the unemployed group. a. Are these percentages (probabilities) ordinary or conditional? Explain, by specifying events to which the probabilities refer. b. Of the individuals polled, 1266 were unemployed and 400 were actively disengaged. Create a contingency table showing counts for job status and self- classification. c. Create a tree diagram such that the first branching represents job status and the second branching represent self-classification. Be sure to include the appropriate percentages on each branch.

An article \(^{4}\) in www.networkworld .com about evaluating e-mail filters that are designed to detect spam described a test of MailFrontier's Anti-Spam Gateway (ASG). In the test, there were 7840 spam messages, of which ASG caught 7005 . Of the 7053 messages that ASG identified as spam, they were correct in all but 48 cases. a. Set up a contingency table that cross classifies the actual spam status (with the rows "spam" and "not spam") by the ASG filter prediction (with the columns "predict message is spam" and "predict message is not spam"). Using the information given, enter counts in three of the four cells. b. For this test, given that a message is truly spam, estimate the probability that ASG correctly detects it. c. Given that ASG identifies a message as spam, estimate the probability that the message truly was spam.

Because of the increasing nuisance of spam e-mail messages, many start-up companies have emerged to develop e-mail filters. One such filter was recently advertised as being \(95 \%\) accurate. The way the advertisement is worded, \(95 \%\) accurate could mean that (a) \(95 \%\) of spam is blocked, (b) \(95 \%\) of valid e-mail is allowed through, (c) \(95 \%\) of the e-mail allowed through is valid, or (d) \(95 \%\) of the blocked e-mail is spam. Let S denote \\{message is spam\\}, and let \(\mathrm{B}\) denote \(\\{\) filter blocks message \(\\} .\) Using these events and their complements, identify each of these four possibilities as a conditional probability.

A pollster wants to estimate the proportion of Canadian adults who support the prime minister's performance on the job. He comments that by the law of large numbers, to ensure a sample survey's accuracy, he does not need to worry about the method for selecting the sample, only that the sample has a very large sample size. Do you agree with the pollster's comment? Explain.

A local downtown arts and crafts shop found from past observation that \(20 \%\) of the people who enter the shop actually buy something. Three potential customers enter the shop. a. How many outcomes are possible for whether the clerk makes a sale to each customer? Construct a tree diagram to show the possible outcomes. \((\) Let \(Y=\) sale \(, N=\) no sale. \()\) b. Find the probability of at least one sale to the three customers. c. What did your calculations assume in part b? Describe a situation in which that assumption would be unrealistic.
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