/*! 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 13 Suppose that events \(E\) and \(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 13

Suppose that events \(E\) and \(F\) are independent, \(P(E)=0.8\), and \(P(F)=0.5 .\) What is \(P(E\) and \(F) ?\)

Short Answer

Expert verified
0.4

Step by step solution

01

- Understand Independence

When two events are independent, the occurrence of one event does not affect the occurrence of the other. This means the probability of both events happening together is the product of their individual probabilities.
02

- Recall the Formula for Independent Events

For independent events, the probability of both events occurring, denoted as \(P(E \cap F)\), is given by \[P(E \cap F) = P(E) \times P(F).\]
03

- Plug in the Given Probabilities

Substitute the given values into the formula: \[P(E \cap F) = 0.8 \times 0.5.\]
04

- Perform the Calculation

Multiply the probabilities: \[0.8 \times 0.5 = 0.4.\]

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.

Independent Events
In probability theory, understanding the concept of independent events is crucial. Independent events are those where the occurrence of one event does not influence the likelihood of the other event happening. For example, flipping a coin and rolling a die are independent events. The result of the coin toss does not affect the outcome of the die roll.
To test for independence between two events, you can check if the probability of both events occurring together equals the product of their individual probabilities. Mathematically, events E and F are independent if \(P(E \cap F) = P(E) \times P(F).\)
Recognizing this property will help you to simplify complex problems involving multiple events.
Probability Calculation
Calculating probability involves determining the likelihood of an event happening out of all possible outcomes. In this context, the probability of an event E, denoted as P(E), is calculated using the formula:

  • \( P(E) = \frac{Number \ of \ favourable \ outcomes}{Total \ number \ of \ possible \ outcomes} \)

In problems with independent events, you'll often need to handle multiple steps, such as calculating individual probabilities first and then using these to find the joint probability. Remember that probabilities are always between 0 and 1, where 0 means an event is impossible, and 1 means it's certain to happen.
Exercise helps in practicing these calculations, ensuring you can confidently determine probabilities in various scenarios.
Multiplication Rule for Independent Events
The multiplication rule is essential when dealing with independent events. It states that for two independent events E and F, the probability that both events occur is the product of their individual probabilities.
To apply this rule, follow these steps:

  • Identify that the events are independent
  • Use the formula for the probability of both events: \(P(E \cap F) = P(E) \times P(F)\)
  • Substitute the given probabilities into the formula

For example, if you know that \(P(E) = 0.8\) and \(P(F) = 0.5\), you can calculate \(P(E \cap F)\) as:
\[P(E \cap F) = 0.8 \times 0.5 = 0.4.\]
This rule is immensely powerful in breaking down seemingly complex problems into more manageable calculations.

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

Companies whose stocks are listed on the New York Stock Exchange (NYSE) have their company name represented by either one, two, or three letters (repetition of letters is allowed). What is the maximum number of companies that can be listed on the New York Stock Exchange?

A box containing twelve 40-watt light bulbs and eighteen 60-watt light bulbs is stored in your basement. Unfortunately, the box is stored in the dark and you need two 60-watt bulbs. What is the probability of randomly selecting two 60-watt bulbs from the box?

In 1991 , columnist Marilyn Vos Savant posted her reply to a reader's question. The question posed was in reference to one of the games played on the gameshow Let's Make a Deal hosted by Monty Hall. Her reply generated a tremendous amount of backlash, with many highly educated individuals angrily responding that she was clearly mistaken in her reasoning. (a) Using subjective probability, estimate the probability of winning if you switch. (b) Load the Let's Make a Deal applet at www.pearsonhighered.com/sullivanstats. Simulate the probability that you will win if you switch by going through the simulation at least 100 times. How does your simulated result compare to your answer to part (a)? (c) Research the Monty Hall Problem as well as the reply by Marilyn Vos Savant. How does the probability she gives compare to the two estimates you obtained? (d) Write a report detailing why Marilyn was correct. One approach is to use a random variable on a wheel similar to the one shown. On the wheel, the innermost ring indicates the door where the car is located, the middle ring indicates the door you selected, and the outer ring indicates the door(s) that Monty could show you. In the outer ring, green indicates you lose if you switch while purple indicates you win if you switch.

A salesperson must travel to eight cities to promote a new marketing campaign. How many different trips are possible if any route between cities is possible?

Adult Americans (18 years or older) were asked whether they used social media (Facebook, Twitter, and so on ) regularly. The following table is based on the results of the survey. $$ \begin{array}{lccccc} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \begin{array}{l} \text { Use social } \\ \text { media } \end{array} & 117 & 89 & 83 & 49 & \mathbf{3 3 8} \\ \hline \begin{array}{l} \text { Do not use } \\ \text { social media } \end{array} & 33 & 36 & 57 & 66 & \mathbf{1 9 2} \\ \hline \text { Total } & \mathbf{1 5 0} & \mathbf{1 2 5} & \mathbf{1 4 0} & \mathbf{1 1 5} & \mathbf{5 3 0} \\ \hline \end{array} $$ (a) What is the probability that a randomly selected adult American uses social media, given the individual is \(18-34\) years of age? (b) What is the probability that a randomly selected adult American is \(18-34\) years of age, given the individual uses social media? (c) Are 18 - to 34 -year olds more likely to use social media than individuals in general? Why?
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Applied Mathematics

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.