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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 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 \(S\).

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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.

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Most popular questions from this chapter

Thirty-two persons have applied for a security guard position with a company. Of them, 7 have previous experience in this area and 25 do not. Suppose one applicant is selected at random. Consider the following two events: This applicant has previous experience, and this applicant does not have previous experience. If you are to find the probabilities of these two events, would you use the classical approach or the relative frequency approach? Explain why.

Many states have a lottery game, usually called a Pick-4, in which you pick a four-digit number such as 7359 . During the lottery drawing, there are four bins, each containing balls numbered 0 through 9\. One ball is drawn from each bin to form the four-digit winning number. a. You purchase one ticket with one four-digit number. What is the probability that you will win this lottery game? b. There are many variations of this game. The primary variation allows you to win if the four digits in your number are selected in any order as long as they are the same four digits as obtained by the lottery agency. For example, if you pick four digits making the number 1265, then you will win if \(1265,2615,5216,6521\), and so forth, are drawn. The variations of the lottery game depend on how many unique digits are in your number. Consider the following four different versions of this game. i. All four digits are unique (e.g., 1234 ) ii. Exactly one of the digits appears twice (e.g., 1223 or 9095 ) iii. Two digits each appear twice (e.g., 2121 or 5588 ) iv. One digit appears three times (e.g., 3335 or 2722 ) Find the probability that you will win this lottery in each of these four situations.

Powerball is a game of chance that has generated intense interest because of its large jackpots. To play this game, a player selects five different numbers from 1 through 59 , and then picks a Powerball number from 1 through 39 . The lottery organization randomly draws 5 different white balls from 59 balls numbered 1 through 59 , and then randomly picks a Powerball number from 1 through \(39 .\) Note that it is possible for the Powerball number to be the same as one of the first five numbers. a. If the player's first five numbers match the numbers on the five white balls drawn by the lottery organization and the player's Powerball number matches the Powerball number drawn by the lottery organization, the player wins the jackpot. Find the probability that a player who buys one ticket will win the jackpot. (Note that the order in which the five white balls are drawn is unimportant.) b. If the player's first five numbers match the numbers on the five white balls drawn by the lottery organization, the player wins about \(\$ 200,000\). Find the probability that a player who buys one ticket will win this prize.

A statistical experiment has eight equally likely outcomes that are denoted by \(1,2,3,4,5,6,7\), and 8\. Let event \(A=\\{2,5,7\\}\) and event \(B=\\{2,4,8\\}\). a. Are events \(A\) and \(B\) mutually exclusive events? b. Are events \(A\) and \(B\) independent events? c. What are the complements of events \(A\) and \(B\), respectively, and their probabilities?

Two thousand randomly selected adults were asked if they think they are financially better off than their parents. The following table gives the two- way classification of the responses based on the education levels of the persons included in the survey and whether they are financially better off, the same as. or worse off than their parents. $$\begin{array}{lccc} \hline & \begin{array}{c} \text { Less Than } \\ \text { High School } \end{array} & \begin{array}{c} \text { High } \\ \text { School } \end{array} & \begin{array}{c} \text { More Than } \\ \text { High School } \end{array} \\ \hline \text { Better off } & 140 & 450 & 420 \\ \text { Same as } & 60 & 250 & 110 \\ \text { Worse off } & 200 & 300 & 70 \\ \hline \end{array}$$ a. Suppose one adult is selected at random from these 2000 adults. Find the following probabilities. i. \(P\) (better off and high school) ii. \(P(\) more than high school and worse off ) b. Find the joint probability of the events "worse off" and "better off." Is this probability zero? Explain why or why not.
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