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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

An appliance repair company that makes service calls to customers' homes has found that \(5 \%\) of the time there is nothing wrong with the appliance and the problem is due to customer error (appliance unplugged, controls improperly set, etc.). Two service calls are selected at random, and it is observed whether or not the problem is due to customer error. Draw a tree diagram. Find the probability that in this sample of two service calls a. both problems are due to customer error b. at least one problem is not due to customer error

A gambler has four cards - two diamonds and two clubs. The gambler proposes the following game to you: You will leave the room and the gambler will put the cards face down on a table. When you return to the room, you will pick two cards at random. You will win $$\$ 10$$ if both cards are diamonds, you will win $$\$ 10$$ if both are clubs, and for any other outcome you will lose $$\$ 10$$. Assuming that there is no cheating, should you accept this proposition? Support your answer by calculating your probability of winning $$\$ 10$$.

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.

Recent uncertain economic conditions have forced many people to change their spending habits. In a recent telephone poll of 1000 adults, 629 stated that they were cutting back on their daily spending. Suppose that 322 of the 629 people who stated that they were cutting back on their daily spending said that they were cutting back "somewhat" and 97 stated that they were cutting back "somewhat" and "delaying the purchase of a new car by at least 6 months". If one of the 629 people who are cutting back on their spending is selected at random, what is the probability that he/she is delaying the purchase of a new car by at least 6 months given that he/she is cutting back on spending "somewhat?"

Explain the meaning of the union of two events. Give one example.
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