/*! 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 2 What is the probability of an ev... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 2

What is the probability of an event that is impossible? Suppose a probability is approximated to be zero based on empirical results. Does this mean the event is impossible?

Short Answer

Expert verified
The probability of an impossible event is exactly 0. However, a probability approximated to zero does not mean the event is impossible.

Step by step solution

01

Understand the Definition of an Impossible Event

An impossible event in probability is an event that cannot happen under any circumstances. Its probability is zero.
02

Identify the Probability of an Impossible Event

Since an impossible event cannot occur, its probability is mathematically defined as 0. This can be written as: \( P(\text{Impossible Event}) = 0 \)
03

Determine If a Probability Approximated to Zero Means Impossibility

A probability approximated to be zero based on empirical results does not necessarily mean the event is impossible. It indicates that it is highly unlikely, but there is still a minuscule chance of the event occurring.

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.

Empirical Probability
Empirical probability refers to the likelihood of an event based on observed data or experimentation. Instead of theoretical calculations, we rely on past outcomes to estimate how probable an event is. For instance, if you observe it raining 15 out of 100 days, then the empirical probability of rain on any given day is calculated as: \( P(\text{Rain}) = \frac{15}{100} = 0.15 \). This method is particularly useful when predicting events where historical data is available. However, keep in mind that empirical probabilities can change as more data is collected, providing a more accurate representation of the true likelihood.

To summarize:
  • Empirical Probability is based on actual data.
  • It might change with the addition of new information.
  • It provides an evidence-based likelihood of an event.
Impossible Events
An event is classified as impossible when there is no possible way for it to happen under given circumstances. In probability theory, these events have a probability of zero. For example, rolling a seven on a standard six-sided die is an impossible event since six is the highest number possible on the die. Thus, we can state: \( P(\text{Rolling a seven}) = 0 \).

While this is clear mathematically, it is essential to differentiate between an event being impossible and being extremely unlikely. Just because something has a very low probability, close to zero, does not mean it is impossible.
  • Impossible Events can't occur by any means.
  • Probability of impossible events is exactly zero.
  • Common misinterpretations can confuse 'impossible' with 'highly unlikely'.'
Mathematical Probability
Mathematical probability provides a theoretical framework for predicting the likelihood of events. It uses mathematical formulas and principles rather than observation. For example, the probability of flipping a fair coin and it landing on heads is: \( P(\text{Heads}) = \frac{1}{2} \) since there are two possible outcomes, heads or tails, each equally likely. This is based on the assumption that the coin is fair, and each flip is independent of the others.

Mathematical probability is powerful because it gives precise and objective measures for predicting events. It's necessary to understand these probabilities to make informed decisions based on statistics.
  • Mathematical Probability relies on theoretical calculations.
  • It often involves fair assumptions for models.
  • Helps predict outcomes with a high level of accuracy.

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

The probability that a randomly selected individual in the United States 25 years and older has at least a bachelor's degree is \(0.272 .\) The probability that an individual in the United States 25 years and older has at least a bachelor's degree, given that the individual is Hispanic, is 0.114. Are the events "bachelor's degree" and "Hispanic" independent? (Source: Educational Attainment in the United States, 2003. U.S. Census Bureau, June 2004 )

A baseball team consists of three outfielders, four infielders, a pitcher, and a catcher. Assuming that the outfielders and infielders are indistinguishable, how many batting orders are possible?

Suppose a single card is selected from a standard 52-card deck. What is the probability that the card drawn is a king? Now suppose a single card is drawn from a standard 52-card deck, but we are told that the card is a heart. What is the probability that the card drawn is a king? Did the knowledge that the card is a heart change the probability that the card was a king? What is the term used to describe this result?

In the Big Game, an urn contains balls numbered 1 to \(50,\) and a second urn contains balls numbered 1 to \(36 .\) From the first urn, 5 balls are chosen randomly, without replacement. From the second urn, 1 ball is chosen randomly. For a \(\$ 1\) bet, a player chooses one set of five numbers to match the balls selected from the first urn and one number to match the ball selected from the second urn. To win, all six numbers must match; that is, the player must match the first 5 balls selected from the first urn and the single ball selected from the second urn. What is the probability of winning the Big Game with a single ticket?

A probability experiment is conducted in which the sample space of the experiment is, \(S=\\{1,2,3,4,5,6,7,8,9,10,11,12\\} .\) Let event \(E=\\{2,3,4,5,6,7\\},\) event \(F=\\{5,6,7,8,9\\},\) event \(G=\\{9,10,11,12\\},\) and event \(H=\\{2,3,4\\} .\) Assume each outcome is equally likely. List the outcomes in \(F\) or \(G .\) Now find \(P(F \text { or } G)\) by counting the number of outcomes in \(F\) or \(G .\) Determine \(P(F \text { or } G)\) using the General Addition Rule.
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Geometry

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.