/*! 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 9 Draw a tree diagram for three to... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 9

Draw a tree diagram for three tosses of a coin. List all outcomes for this experiment in a sample space \(S\).

Short Answer

Expert verified
The tree diagram will have eight branches each representing a possible outcome of three coin tosses. The sample space \(S\) will be \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}.

Step by step solution

01

Draw the tree diagram for the first coin toss

Start by drawing a line which branches into two paths. On one path write 'H' representing heads, and on the other write 'T' representing tails. These are the two possible outcomes from the first coin toss.
02

Draw the tree diagram for the second coin toss

For each of the outcomes of the first coin toss, the coin is tossed again. This means for each of the two branches out from the initial line, draw further two branches representing 'H' and 'T'.
03

Draw the tree diagram for the third coin toss

The third coin toss will be represented by two branches (representing 'H' and 'T') coming out from each of the four outcomes from the second coin toss.
04

Identify all outcomes from the tree diagram

The outcomes of the three coin tosses are represented by the paths from the start of the diagram to each of the 'leaves' of the tree. There should be eight potential outcomes.
05

List all outcomes in a sample space \(S\)

The eight possible outcomes are \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}. This set of outcomes represents sample space \(S\).

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.

Understanding Sample Space
In probability theory, the term 'sample space' refers to the set of all possible outcomes of a statistical experiment. When flipping a coin three times, each sequence of heads (H) and tails (T) forms an outcome. For our three coin tosses, the sample space, denoted as \( S \), includes all possible combinations of these outcomes.
You can think of the sample space as a complete list that captures every possibility, ensuring no uncertainty remains about what could possibly occur in the experiment.
  • The sample space for three coin tosses is \( \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \).
  • This means there are eight different sequences in the sample space.
Understanding the sample space is crucial because it provides the foundation for calculating probabilities.
Exploring Probability Outcomes
Probability outcomes refer to the different results that can arise from an experiment. In a three-coin-toss scenario, each outcome forms part of the sample space. The probability of any single outcome occurring is determined by dividing the number of favorable results by the total number of possible results in the sample space.
For coin tosses, each outcome is equally likely since a coin has no preference for heads or tails. Thus, the probability of obtaining any specific sequence, like \( HHH \) or \( THT \), is:
\[ P(\text{outcome}) = \frac{1}{8} \]
  • Every outcome has an equal chance of occurring.
  • Understanding these outcomes helps predict and analyze the results confidently.
Coin Toss Experiment
The coin toss is a classic statistical experiment due to its simplicity and randomness. In our exercise, we examine a sequence of three tosses. Each toss can result in either heads (H) or tails (T), making it ideal for illustrating probability concepts.
Even though a single coin flip appears simple, combining multiple flips increases complexity and invites a broader exploration of outcomes. This unfolding complexity is best visualized using a tree diagram, which maps out all possible outcomes in a structured and understandable way.
  • Tree diagrams help organize information clearly.
  • They visually break down each step of the experiment.
Learning about coin toss experiments reinforces fundamental probability principles, offering a tangible way to see theories in action.
Analyzing Statistical Experiments
Statistical experiments like the three-coin-toss exercise are pivotal in understanding probability. These experiments allow us to observe outcomes, assess probabilities, and reason quantitatively about uncertainty.
By conducting statistical experiments, we learn to gather data effectively, analyze results, and make informed predictions.
  • Each experiment increases our understanding of probability theory.
  • They teach us to apply mathematical frameworks to analyze real-life scenarios.
Understanding the mechanics and results of statistical experiments cultivates analytical skills, enabling learners to transition from theoretical concepts to practical applications.

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

Given that \(A\) and \(B\) are two independent events, find their joint probability for the following. a. \(P(A)=.20\) and \(P(B)=.76\) b. \(P(A)=.57\) and \(P(B)=.32\)

A player plays a roulette game in a casino by betting on a single number each time. Because the wheel has 38 numbers, the probability that the player will win in a single play is \(1 / 38\). Note that each play of the game is independent of all previous plays a. Find the probability that the player will win for the first time on the 10 th play. b. Find the probability that it takes the player more than 50 plays to win for the first time c. The gambler claims that because he has 1 chance in 38 of winning each time he plays, he is certain to win at least once if he plays 38 times. Does this sound reasonable to you? Find the probability that he will win at least once in 38 plays

What is meant by the joint probability of two or more events? Give one example.

Given that \(A\) and \(B\) are two independent events, find their joint probability for the following. a. \(P(A)=.61\) and \(P(B)=.27\) b. \(P(A)=.39\) and \(P(B)=.63\)

According to a March 2009 Gallup Poll (http://www.gallup.com/poll/1 17025/Support-NuclearEnergy-Inches-New-High.aspx), \(71 \%\) of Republicans/Republican leaners and \(52 \%\) of Democrats/Democrat leaners favor the use of nuclear power. The survey consisted of 1012 American adults, approximately half of whom were Republicans or Republican leaners. Suppose the following table gives the distribution of responses of these 1012 adults $$\begin{array}{lcc} \hline & \text { Favor } & \text { Do not favor } \\ \hline \text { Republicans/Republican leaners } & 381 & 128 \\ \text { Democrats/Democrat leaners } & 258 & 245 \\ \hline \end{array}$$ a. If one person is selected at random from this sample of 1012 U.S. adults, find the probability that this person i. does not favor the use of nuclear power ii. is a Republican/Republican leaner iii. favors the use of nuclear power given that the person is a Republican/Republican leaner jv. is a Republican/Republican leaner given that the person does not favor the use of nuclear power b. Are the events favors and does not favor mutually exclusive? What about the events does not favor and Republican/Republican leaner? c. Are the events does not favor and Republican/Republican leaner independent? Why or why not?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.