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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

Define the following two events for two tosses of a coin: \(A=\) at least one head is obtained \(B=\) both tails are obtained a. Are \(A\) and \(B\) mutually exclusive events? Are they independent? Explain why or why not. b. Are \(A\) and \(B\) complementary events? If yes, first calculate the probability of \(B\) and then calculate the probability of \(A\) using the complementary event rule.

Suppose a randomly selected passenger is about to go through the metal detector at JFK Airport in New York City. Consider the following two outcomes: The passenger sets off the metal detector, and the passenger does not set off the metal detector. Are these two outcomes equally likely? Explain why or why not. If you are to find the probability of these two outcomes, would you use the classical approach or the relative frequency approach? Explain why

A gambler has given you two jars and 20 marbles. Of these 20 marbles, 10 are red and 10 are green You must put all 20 marbles in these two jars in such a way that each jar must have at least one marble in it. Then a friend of yours, who is blindfolded, will select one of the two jars at random and then will randomly select a marble from this jar. If the selected marble is red, you and your friend win \(\$ 100\) a. If you put 5 red marbles and 5 green marbles in each jar, what is the probability that your friend selects a red marble? b. If you put 2 red marbles and 2 green marbles in one jar and the remaining marbles in the other jar, what is the probability that your friend selects a red marble? c. How should these 20 marbles be distributed among the two jars in order to give your friend the highest possible probability of selecting a red marble?

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.

A certain state's auto license plates have three letters of the alphabet followed by a three-digit number. a. How many different license plates are possible if all three-letter sequences are permitted and any number from 000 to 999 is allowed? b. Arnold witnessed a hit-and-run accident. He knows that the first letter on the license plate of the offender's car was a \(\mathrm{B}\), that the second letter was an \(\mathrm{O}\) or a \(\mathrm{Q}\), and that the last number was a 5. How many of this state's license plates fit this description?
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