/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Describe the sample space \(S\) ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 12

Describe the sample space \(S\) of the experiment and list the elements of the given event. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) HINT [See Examples 1-3.] A letter is chosen at random from those in the word Mozart; the letter is neither \(a\) nor \(m\).

Short Answer

Expert verified
The sample space S in the experiment is all distinct letters in the word "Mozart": \(S = \{M, o, z, a, r, t\}\). The given event is selecting a letter that is neither 'a' nor 'm', so the elements of the given event E are: \(E = \{o, z, r, t\}\).

Step by step solution

01

List the distinct letters in the word Mozart.

The distinct letters in the word "Mozart" are: \(M\), \(o\), \(z\), \(a\), \(r\), and \(t\).
02

Define the sample space S.

In this experiment, the sample space S includes all 6 distinct letters from the word "Mozart". So, S could be represented as: \(S = \{M, o, z, a, r, t\}\)
03

List the elements of the given event satisfying the criterion.

The given event is selecting a letter from the word "Mozart" that is neither 'a' nor 'm'. So, we need to exclude 'a' and 'M' from the sample space S. The elements in the given event are: \(o\), \(z\), \(r\), and \(t\).
04

Write the elements of the given event in set notation form.

Let's denote the given event by 'E'. In set notation form, the elements of the given event E are: \(E = \{o, z, r, t\}\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Probability Experiment
In statistics and probability, a probability experiment is a process or action that results in distinct outcomes. Each of these outcomes is typically observable and quantifiable. When conducting a probability experiment, we define all possible results to encompass what is known as the sample space.
For example, if we randomly select a letter from the word "Mozart," we are conducting a probability experiment. The experiment's outcome depends on which letter is selected, and the full set of possible outcomes is the sample space.
With each probability experiment, being clear about the parameters and possible results helps us later understand and calculate probabilities.
Set Notation
Set notation is a mathematical way to describe and represent collections of objects or numbers. It is particularly useful in probability because it allows us to clearly define sample spaces and events. Sets are typically written using curly braces. The elements inside these braces represent the different outcomes or members of the set.
For instance, in the word "Mozart," if we list the distinct letters contained in it, we could use set notation to represent them as: \( S = \{ M, o, z, a, r, t \} \)
Here, the set \( S \) represents the sample space of our probability experiment, encompassing every letter that could be selected in the experiment.
Distinct Letters
When working with probability experiments related to letters or words, understanding the concept of distinct letters is important. Distinct letters are the unique characters found within a given word or phrase.
For example, with the word "Mozart," the distinct letters are \( M, o, z, a, r, \) and \( t \). Each one represents a unique possibility within the sample space for any probability experiment involving the selection of letters from "Mozart."
Noticing which letters are distinct is critical, as it influences how we determine both the sample space and the events involving these letters.
Event Description
An event in probability theory is a specific outcome or a group of outcomes from the sample space that share a common characteristic. For instance, when the task is to select a letter from the word "Mozart" that is neither \(a\) nor \(m\), we define the event according to this rule.
For the word "Mozart," where the sample space \(S\) includes \( M, o, z, a, r, t \), the specific event \(E\) can be written in set notation after excluding \(a\) and \(m\) as: \( E = \{ o, z, r, t \} \)
This event description helps us focus only on outcomes that matter for our current problem. It filters the relevant results according to the conditions or criteria given in the problem statement.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Concern the following chart, which shows the way in which a dog moves its facial muscles when torn between the drives of fight and flight. \({ }^{4}\) The "fight" drive increases from left to right; the "fight" drive increases from top to bottom. (Notice that an increase in the "fight" drive causes its upper lip to lift, while an increase in the "flight" drive draws its ears downward.) \(\nabla\) Let \(E\) be the event that the dog's flight drive is the strongest, let \(F\) be the event that the dog's flight drive is weakest, let \(G\) be the event that the dog's fight drive is the strongest, and let \(H\) be the event that the dog's fight drive is weakest. Describe the following events in terms of \(E, F, G\), and \(H\) using the symbols \(\cap, \cup\), and \(^{\prime}\). a. The dog's flight drive is not strongest and its fight drive is weakest. b. The dog's flight drive is strongest or its fight drive is weakest. c. Neither the dog's flight drive nor its fight drive is strongest.

Some of the dice are cleverly weighted so that each of \(2,3,4\) and 5 is twice as likely to come up as 1 is, and 1 and 6 are equally likely.

Suppose two dice (one red, one green) are rolled. Consider the following events. A: the red die shows \(1 ; B:\) the numbers add to \(4 ; C:\) at least one of the numbers is \(1 ;\) and \(D:\) the numbers do not add to 11. Express the given event in symbols and say how many elements it contains. HINT [See Example 5.] The red die shows 1 and the numbers add to \(4 .\)

Marketing A market survey shows that half the owners of Sorey State Boogie Boards became disenchanted with the product and switched to \(C \& T\) Super Professional Boards the next surf season, while the other half remained loyal to Sorey State. On the other hand, three quarters of the \(\mathrm{C\&T}\) Boogie Board users remained loyal to \(\mathrm{C\&T}\), while the rest switched to Sorey State. Set these data up as a Markov transition matrix, and calculate the probability that a Sorey State Board user will be using the same brand two seasons later. [HINT: See Example 1.]

Explain how the addition principle for mutually exclusive events follows from the general addition principle.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.