/*! 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 5 Let the sample space be \(S=\\{1... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 5

Let the sample space be \(S=\\{1,2,3,4,5,6,7,8,9,10\\} .\) Suppose that the outcomes are equally likely. Compute the probability of the event: \(E=\\{1,3,5,10\\}\)

Short Answer

Expert verified
The probability is \(\frac{2}{5}\).

Step by step solution

01

- Understand the Given Sample Space

The sample space provided is \(S = \{1,2,3,4,5,6,7,8,9,10\}\).This represents all possible outcomes.
02

- Identify the Event

The event provided is \(E = \{1,3,5,10\}\).This means we are interested in these specific outcomes within the sample space.
03

- Determine Number of Elements

Count the number of elements in the sample space, \(S\). Thus, \(|S| = 10\).Also, count the elements in the event \(E\), so \(|E| = 4\).
04

- Calculate the Probability

Utilize the probability formula for equally likely outcomes: \[ P(E) = \frac{|E|}{|S|}\].Substitute the counted elements: \[ P(E) = \frac{4}{10} = \frac{2}{5} \]

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.

Sample Space
In probability, the **sample space** of an experiment or random trial is the set of all possible outcomes or results. For example, in the given problem, the sample space is defined as: \( S = \{1,2,3,4,5,6,7,8,9,10\} \). This means there are 10 possible outcomes when conducting the experiment. Understanding the sample space is crucial as it lays the foundation for calculating probabilities. It helps identify all the possible outcomes and ensures none are overlooked.
Event
An **event** is a specific set of outcomes within the sample space. These outcomes are the ones we are particularly interested in. For instance, in the problem, the event is given by: \( E = \{1,3,5,10\} \). This means we are focused on the outcomes where 1, 3, 5, or 10 occur. An event might consist of a single outcome or multiple outcomes. Understanding what constitutes an event is essential because it helps in determining the probability of its occurrence by comparing it to the sample space.
Equally Likely Outcomes
When we say outcomes are **equally likely**, we mean that each outcome has the same chance of happening. In the context of our problem, every number in the sample space \( S = \{1,2,3,4,5,6,7,8,9,10\} \) is equally likely to occur. This makes our calculation straightforward, as each has an equal probability of \( \frac{1}{10} \). The concept of equally likely outcomes is fundamental in probability because it simplifies the calculation of probabilities by allowing us to divide the number of favorable outcomes by the total number of possible outcomes.
Counting Elements in a Set
To determine probabilities, we often need to count the number of elements in different sets. In our problem, we count the elements in both the sample space and the event. The sample space \( S \) has 10 elements, as shown: \( |S| = 10 \). The event \( E \) has 4 elements: \( |E| = 4 \). This counting allows us to use the probability formula: \( P(E) = \frac{|E|}{|S|} \). By substituting the counts, we get: \( P(E) = \frac{4}{10} = \frac{2}{5} \). Counting elements is a powerful tool in probability as it helps bridge the abstract concepts with concrete numbers.

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

Christmas lights are often designed with a series circuit. This means that when one light burns out the entire string of lights goes black. Suppose that the lights are designed so that the probability a bulb will last 2 years is \(0.995 .\) The success or failure of a bulb is independent of the success or failure of other bulbs. (a) What is the probability that in a string of 100 lights all 100 will last 2 years? (b) What is the probability that at least one bulb will burn out in 2 years?

A family has six children. If this family has exactly two boys, how many different birth and gender orders are possible?

In the game Text Twist, six letters are given and the player must form words of varying lengths using the letters provided. Suppose that the letters in a particular game are ENHSIC. (a) How many different arrangements are possible using all 6 letters? (b) How many different arrangements are possible using only 4 letters? (c) The solution to this game has three 6 -letter words. To advance to the next round, the player needs at least one of the six-letter words. If the player simply guesses, what is the probability that he or she will get one of the six-letter words on their first guess of six letters?

About \(13 \%\) of the population is left-handed. If two people are randomly selected, what is the probability that both are left-handed? What is the probability that at least one is right-handed?

Suppose 40 cars start at the Indianapolis 500. In how many ways can the top 3 cars finish the race?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

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.