/*! 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 4 What does it mean when two event... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 4

What does it mean when two events are complements?

Short Answer

Expert verified
Two events are complements when one event happens if and only if the other does not, and their probabilities add up to 1.

Step by step solution

01

Understanding The Concept of Complements

Two events are said to be complements when one event occurs if and only if the other event does not occur. In other words, they are mutually exclusive and exhaustive. The occurrence of one event completely rules out the occurrence of the other.
02

Defining The Events

Let鈥檚 denote two events as Event A and Event B. If these two events are complements of each other, then Event A happening means that Event B cannot happen, and vice versa.
03

Example Clarification

Consider the example of flipping a coin. Let Event A be getting heads and Event B be getting tails. These two are complements because if you get heads (Event A), you couldn鈥檛 possibly get tails (Event B) in the same flip, and vice versa. The probability of getting heads plus the probability of getting tails equals 1.

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.

Mutually Exclusive Events
Mutually exclusive events are situations where the occurrence of one event means the other event cannot happen at the same time. For instance, when flipping a coin, getting a 'heads' and 'tails' are mutually exclusive because both cannot occur simultaneously.
The key points of mutually exclusive events are:
  • No overlap between the events; one excludes the other.
  • The sum of probabilities for all mutually exclusive events is 1.

Mathematically, if Event A and Event B are mutually exclusive, then
\( P(A \text{ and } B) = 0 \).
The general formula is:
\( P(A \text { or } B) = P(A) + P(B) \).
This is because the probability of either event happening is the sum of their individual probabilities.
Exhaustive Events
Exhaustive events are collections of all possible outcomes of an experiment. This means at least one of these events must happen. Let's continue with the coin flip example: the exhaustive events when flipping a coin are getting 'heads' or 'tails.' Together, these cover all possible outcomes.
  • Every possible outcome is accounted for.
  • The sum of probabilities of exhaustive events is always 1.

If events are both mutually exclusive and exhaustive, the probability of one of the events occurring is:
\( P(A) + P(B) + P(C) + \text {etc.} = 1 \).
In simple terms, you're certain that one of the events will occur, and there are no other possible outcomes.
Probability Theory
Probability theory is the branch of mathematics that deals with calculating the likelihood of a given event's occurrence. It provides a way to quantify uncertainty. The probability of an event ranges from 0 to 1:
  • A probability of 0 means the event will not happen.
  • A probability of 1 means the event will certainly happen.
  • Any probability between 0 and 1 indicates how likely the event is to happen.

The probability of an event A is often written as \( P(A) \). Basic probability rules include:
\( P(A) + P(\text {not } A) = 1 \).
This rule indicates that the probability of all possible outcomes of an event sums up to 1.
Additionally:
\( P(A \text { or } B) = P(A) + P(B) - P(A \text { and } B) \).
This formula represents the probability of either event A or event B happening, accounting for any overlap between them.

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

Suppose a shipment of 120 electronic components contains 4 defective components. To determine whether the shipment should be accepted, a quality- control engineer randomly selects 4 of the components and tests them. If 1 or more of the components is defective, the shipment is rejected. What is the probability the shipment is rejected?

Suppose you have just received a shipment of 100 televisions. Although you don't know this, 6 are defective. To determine whether you will accept the shipment, you randomly select 5 televisions and test them. If all 5 televisions work, you accept the shipment; otherwise, the shipment is rejected. What is the probability of accepting the shipment?

The notation \(P(F | E)\) means the probability of event _________ given event _________ .

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

Among 21 - to 25 -year-olds, \(29 \%\) say they have driven while under the influence of alcohol. Suppose three \(21-\) to 25 -year-olds are selected at random. Source: U.S. Department of Health and Human Services, reported in USA Today (a) What is the probability that all three have driven while under the influence of alcohol? (b) What is the probability that at least one has not driven while under the influence of alcohol? (c) What is the probability that none of the three has driven while under the influence of alcohol? (d) What is the probability that at least one has driven while under the influence of alcohol?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Applied 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.