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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

The probability that a corporation makes charitable contributions is .72. Two corporations are selected at random, and it is noted whether or not they make charitable contributions. a. Draw a tree diagram for this experiment. b. Find the probability that at most one corporation makes charitable contributions.

The probability that a farmer is in debt is 80 . What is the probability that three randomly selected farmers are all in debt? Assume independence of events.

A screening test for a certain disease is prone to giving false positives or false negatives. If a patient being tested has the disease, the probability that the test indicates a (false) negative is \(.13 .\) If the patient does not have the disease, the probability that the test indicates a (false) positive is .10. Assume that \(3 \%\) of the patients being tested actually have the disease. Suppose that one patient is chosen at random and tested. Find the probability that a. this patient has the disease and tests positive b. this patient does not have the disease and tests positive c. this patient tests positive d. this patient has the disease given that he or she tests positive (Hint: A tree diagram may be helpful in part c.)

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

Five hundred employees were selected from a city's large private companies, and they were asked whether or not they have any retirement benefits provided by their companies. Based on this information, the following two-way classification table was prepared. $$ \begin{array}{lcc} & {\text { Have Retirement Benefits }} \\ \hline & \text { Yes } & \text { No } \\ \hline \text { Men } & 225 & 75 \\ \text { Women } & 150 & 50 \\ \hline \end{array} $$ Suppose one employee is selected at random from these 500 employees. Find the following probabilities. a. The probability of the union of events "woman" and "yes" b. The probability of the union of events "no" and "man"
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