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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 would show eight endpoints, each representing a different outcome of three coin tosses. The sample space \(S\) containing all outcomes is \(S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\)

Step by step solution

01

Set up the Tree Diagram Structure

A tree diagram for tossing a coin three times splits into two branches at each stage for each possible resultn; firstly for the first coin toss, then for the second and finally for the third. This leads to eight possible outcomes, represented as endpoints of the branches.
02

Label the Tree Diagram

Label the first set of branches as 'first toss', the second as 'second toss', and the third as 'third toss'. On each branch, assign the possible outcome of the toss; either heads (H) or tails (T). Each sequence from the root of tree to the leaves represents a unique outcome of the three coin tosses.
03

List Outcomes in Sample Space S

The sample space \(S\) is the set of all possible outcomes. As each final branch on the tree represents an outcome, list out each combination to form the sample space. This gives \(S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\).

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Key Concepts

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

Tree Diagram
A tree diagram is a visual tool used to map out and explore all possible outcomes of a probability experiment.
In the case of coin tossing, it helps represent each possible result at every stage of the experiment.
  • Begin your tree diagram with a single starting point, known as the root.
  • Each possible outcome of a coin toss, heads (H) or tails (T), branches out from the root.
  • For a sequence of three coin tosses, continue to branch each outcome into two more paths for the next toss. Repeat this for all three tosses.
  • Your diagram will have 8 endpoints, each representing a distinct outcome after three tosses.
This branching visually demonstrates how outcomes build upon each other and helps make sense of complex probability problems.
Sample Space
In probability, the sample space is the set of all possible outcomes for a given experiment.
For three tosses of a coin, the sample space consists of all possible combinations of heads and tails.
  • The sample space is typically denoted by the letter \( S \).
  • For our experiment, it includes: \( S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\} \).
  • Each element of the sample space corresponds to a final outcome represented in the tree diagram.
This set forms the basis for calculating probabilities, since each outcome is equally likely when tossing a fair coin.
Coin Toss
A coin toss is a basic probability experiment involving flipping a coin and observing whether it lands on heads or tails.
It is often used in examples due to its simplicity and predictable set of outcomes.
  • Each toss of the coin has exactly two equally likely outcomes: heads (H) or tails (T).
  • The simplicity makes it easier to see how probabilities work for multiple events combined.
  • Understanding the probabilities of coin tosses is foundational to grasping more complex scenarios in probability theory.
This foundational concept helps in learning how independent events (like multiple coin tosses) combine to form an overall probability model.
Outcomes
Outcomes are the results of a probability experiment.
Each sequence in the tree diagram corresponds to one outcome.
  • For our three-toss experiment, there are eight possible outcomes.
  • These include scenarios like all heads (\(HHH\)) and all tails (\(TTT\)), among others.
  • Each outcome is equally likely when the coin is fair, making it an essential concept for probability.
Contemplating the possible outcomes helps in understanding the sample space and calculating probabilities for each potential result.

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Most popular questions from this chapter

How is the addition rule of probability for two mutually exclusive events different from the rule for two mutually nonexclusive events?

How is the multiplication rule of probability for two dependent events different from the rule for two independent events?

In a group of people, some are in favor of a tax increase on rich people to reduce the federal deficit and others are against it. (Assume that there is no other outcome such as "no opinion" and "do not know.") Three persons are selected at random from this group and their opinions in favor or against raising such taxes are noted. How many total outcomes are possible? Write these outcomes in a sample space \(S\). Draw a tree diagram for this experiment.

A box contains three items that are labeled \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\). Two items are selected at random (without replacement) from this box. List all the possible outcomes for this experiment. Write the sample space \(S .\)

A random sample of 250 juniors majoring in psychology or communication at a large university is selected. These students are asked whether or not they are happy with their majors. The following table gives the results of the survey. Assume that none of these 250 students is majoring in both areas. $$ \begin{array}{lcc} \hline & \text { Happy } & \text { Unhappy } \\ \hline \text { Psychology } & 80 & 20 \\ \text { Communication } & 115 & 35 \\ \hline \end{array} $$ a. If one student is selected at random from this group, find the probability that this student is i. happy with the choice of major ii. a psychology major iii. a communication major given that the student is happy with the choice of major iv. unhappy with the choice of major given that the student is a psychology major v. a psychology major and is happy with that major vi. a communication major \(o r\) is unhappy with his or her major b. Are the events "psychology major" and "happy with major" independent? Are they mutually exclusive? Explain why or why not.
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