/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 102 Again Suppose a friend says he c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 102

Again Suppose a friend says he can predict whether a coin flip will result in heads or tails. You test him, and he gets 20 right out of \(20 .\) Do you think he can predict the coin flip (or has a way of cheating)? Or could this just be something that is likely to occur by chance? Explain without performing any calculation.

Short Answer

Expert verified
It's highly unlikely that your friend correctly predicted 20 consecutive coin flips simply by chance, meaning they probably have a method of cheating or had incredible luck.

Step by step solution

01

Analyze the Probability of Predicting Correctly Once

In a fair coin flip, there is a 50% chance it will land on heads and a 50% chance it will land on tails, meaning any guess has a 50% chance of being correct.
02

Appreciate the Difficulty of Repeated Success

Even though the likelihood of predicting a coin flip correctly once is quite high (50%), the chances of predicting it correctly multiple times in a row decrease dramatically each time. This is due to the independent nature of each flip - the result of one flip has no impact on the result of the next.
03

Conclusion Drawn Regarding Probability

Predicting the result of 20 coin flips in a row correctly is extraordinarily unlikely to happen purely by chance. The odds of guessing correctly each time are slim. Thus, it could be assumed that the friend either has a way of cheating or an extraordinary amount of good luck.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Coin Flip
A coin flip is one of the simplest forms of probability exercises, often used to illustrate basic concepts of chance and random events. When you flip a coin, there are two possible outcomes: heads or tails. Each outcome is equally likely, assuming the coin is fair.

Understanding a coin flip involves recognizing that it represents a classic probability case where the likelihood of each result is 50%. This simplicity makes it a useful starting point for grasping probability. However, while individual flips are straightforward, predicting a sequence, like getting heads 20 times in a row, becomes complex because of how probabilities multiply.
Independent Events
In the context of probability, independent events are those whose outcomes do not affect one another. This means the result of one event, such as a coin flip, doesn't influence the result of the next.

When flipping a coin, each flip is an independent event. The outcome of flipping heads or tails is not influenced by previous flips. This independence is crucial in understanding probability over several trials. For instance, getting a head on the third flip does not make it more or less likely to get a head on the fourth flip. Each flip stands alone in probability terms.
  • Each event has no memory of the previous events.
  • The probability remains consistent across trials.
  • This independence contributes to the unpredictability of outcomes in a series of coin flips.
Chance
The concept of chance is central to probability and reflects the likelihood of a particular outcome occurring. In any single flip of a coin, the chance of getting heads is 50%, just as the chance of getting tails is 50%.

When you extend this idea to multiple coin flips, calculating the chance of a specific sequence, like predicting all flips correctly, becomes intricate. As each flip is independent, predicting 20 correct outcomes in a row has a chance of \[ \left( \frac{1}{2} \right)^{20} \], which is less than one in a million.
  • Single event chance remains at 50% for each flip.
  • Multiple consecutive events drastically decrease overall likelihood.
  • In practical terms, achieving a long sequence by chance is extremely improbable.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If we reject the null hypothesis, can we claim to have proved that the null hypothesis is false? Why or why not?

A teacher giving a true/false test wants to make sure her students do better than they would if they were simply guessing, so she forms a hypothesis to test this. Her null hypothesis is that a student will get \(50 \%\) of the questions on the exam correct. The alternative hypothesis is that the student is not guessing and should get more than \(50 \%\) in the long run. $$ \begin{aligned} &\mathrm{H}_{0}: p=0.50 \\ &\mathrm{H}_{\mathrm{a}}: p>0.50 \end{aligned} $$ A student gets 30 out of 50 questions, or \(60 \%\), correct. The p-value is \(0.079 .\) Explain the meaning of the \(\mathrm{p}\) -value in the context of this question.

A college chemistry instructor thinks the use of embedded tutors will improve the success rate in introductory chemistry courses. The passing rate for introductory chemistry is \(62 \%\). During one semester, 200 students were enrolled in introductory chemistry courses with an embedded tutor. Of these 200 students, 140 passed the course. a. What is \(\hat{p}\), the sample proportion of students who passed introductory chemistry. b. What is \(p_{0}\), the proportion of students who pass introductory chemistry if the null hypothesis is true? c. Find the value of the test statistic. Explain the test statistic in context.

The mother of a teenager has heard a claim that \(25 \%\) of teenagers who drive and use a cell phone reported texting while driving. She thinks that this rate is too high and wants to test the hypothesis that fewer than \(25 \%\) of these drivers have texted while driving. Her alternative hypothesis is that the percentage of teenagers who have texted when driving is less than \(25 \%\). $$ \begin{aligned} &\mathrm{H}_{0}: p=0.25 \\ &\mathrm{H}_{\mathrm{a}}: p<0.25 \end{aligned} $$ She polls 40 randomly selected teenagers, and 5 of them report having texted while driving, a proportion of \(0.125 .\) The p-value is \(0.034\). Explain the meaning of the \(\mathrm{p}\) -value in the context of this question.

Choosing a Test and Naming the Population(s) In each case, choose whether the appropriate test is a one-proportion z-test or a two-proportion z-test. Name the population(s). a. A researcher takes a random sample of 4 -year-olds to find out whether girls or boys are more likely to know the alphabet. b. A pollster takes a random sample of all U.S. adult voters to see whether more than \(50 \%\) approve of the performance of the current U.S. president. c. A researcher wants to know whether a new heart medicine reduces the rate of heart attacks compared to an old medicine. d. A pollster takes a poll in Wyoming about homeschooling to find out whether the approval rate for men is equal to the approval rate for women. e. A person is studied to see whether he or she can predict the results of coin flips better than chance alone.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.