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Chapter 4: Problem 2

List the simple events for each of the following statistical experiments in a sample space \(S .\) a. One roll of a die \(\quad\) b. Three tosses of a coin \(c\). One toss of a coin and one roll of a die

Short Answer

Expert verified
a. Simple events for a roll of die are: 1, 2, 3, 4, 5, 6. b. For three coin tosses: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. c. For a hybrid experiment of a coin toss and a roll of die, the simple events are: (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6).

Step by step solution

01

Enumerate simple events (Roll of a die)

A die has six faces, each showing a distinct number from 1 to 6. So, when a die is rolled once, the simple events would be the possible outputs: 1, 2, 3, 4, 5 and 6.
02

Enumerate simple events (Three coin tosses)

Simple events for the occurrence of three coin tosses can be enumerated using a binary count pattern. There are 8 possible outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. Where H stands for head and T stands for tail.
03

Enumerate simple events (One toss of a coin and one roll of a die)

This combined experiment can give rise to a total of 12 simple events. They are: (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6).

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Key Concepts

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

Simple Events
Simple events are the building blocks of probability theory and form the foundation of sample spaces. A simple event is an outcome or a single possible result of a probability experiment that cannot be further decomposed. For instance, when rolling a die, each face value鈥攍ike 1, 2, 3, 4, 5, or 6鈥攔epresents an individual simple event.
Similarly, in a coin toss, each result (heads or tails) is a simple event. Simple events are crucial because they help in specifying the sample space clearly and enable us to calculate probabilities effectively. Usually, simple events are equally likely, especially in fair games, allowing for straightforward probability calculations.
Probability Experiment
A probability experiment is a process that leads to one or more outcomes from a set of possible outcomes. It's essentially a real-world event that we perform to study the variation and predictability of results. Each time we perform this experiment, it may produce a different outcome.
For example, rolling a die or tossing a coin are classic examples of probability experiments due to their randomness and unpredictability. In these experiments, various simple events occur, which together form the experiment's sample space. By performing these experiments, we can observe the frequency of outcomes and start estimating probabilities based on repeated trials.
Outcomes
Outcomes are the specific results that come from performing a probability experiment. Each possible result is an outcome, and these outcomes collectively form the sample space. For example, if you roll a die, the potential outcomes are the numbers 1 through 6; if you toss a coin, the outcomes are heads or tails.
Outcomes are vital because they help us to understand and list all possibilities of an experiment occurring. Recognizing all possible outcomes ensures we don't miss important elements of probability calculations, which can lead to a clearer understanding of the probability landscape of the experiment.
Enumeration of Events
Enumeration of events is the process of listing all possible simple events within a sample space. This technique is essential in systematically determining and visualizing the potential outcomes of a probability experiment.
For instance, when enumerating events for a die roll, we list the six numbers: 1, 2, 3, 4, 5, and 6. For a more complex experiment like three coin tosses, enumeration involves outlining all eight possible sequences, such as HHH, HHT, and so on. This enumeration helps in ensuring that no outcome is overlooked and facilitates the calculation of probabilities.
  • Enumeration provides clarity and completeness for probability analysis.
  • It acts as a foundation for further probability computations and theoretical advancements.
By ensuring each event is counted, we maintain accuracy in our statistical evaluations.

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Most popular questions from this chapter

There are a total of 160 practicing physicians in a city. Of them, 75 are female and 25 are pediatricians. Of the 75 females, 20 are pediatricians. Are the events "female" and "pediatrician" independent? Are they mutually exclusive? Explain why or why not.

A thief has stolen Roger's automatic teller machine (ATM) card. The card has a four-digit personal identification number (PIN). The thief knows that the first two digits are 3 and 5 , but he does not know the last two digits. Thus, the PIN could be any number from 3500 to 3599 . To protect the customer, the automatic teller machine will not allow more than three unsuccessful attempts to enter the PIN. After the third wrong PIN, the machine keeps the card and allows no further attempts. a. What is the probability that the thief will find the correct PIN within three tries? (Assume that the thief will not try the same wrong PIN twice.) b. If the thief knew that the first two digits were 3 and 5 and that the third digit was either 1 or 7 , what is the probability of the thief guessing the correct PIN in three attempts?

The probability that a randomly selected college student attended at least one major league baseball game last year is .12. What is the complementary event? What is the probability of this complementary event?

Let \(A\) be the event that a number less than 3 is obtained if we roll a die once. What is the probability of \(A ?\) What is the complementary event of \(A\), and what is its probability?

Given that \(P(B)=.65\) and \(P(A\) and \(B)=.45\), find \(P(A \mid B)\).
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