/*! 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 8 If \(P(E)=0.9\) and \(P(F)=0.8\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 8

If \(P(E)=0.9\) and \(P(F)=0.8\), show that \(P(E F) \geqslant 0.7\). In general, show that $$ P(E F) \geqslant P(E)+P(F)-1 $$ This is known as Bonferroni's inequality.

Short Answer

Expert verified
In general, Bonferroni's inequality states that \( P(E 鈭� F) \geq P(E) + P(F) - 1 \). Given the probabilities \( P(E) = 0.9 \) and \( P(F) = 0.8 \), by applying this inequality, we find that \( P(E 鈭� F) \geq 0.9 + 0.8 - 1 = 0.7 \). Hence, we have shown that \( P(E 鈭� F) \geq 0.7 \), verifying the truth of Bonferroni's inequality in this case.

Step by step solution

01

Understand the problem

We need to show that the probability of the intersection of two events is greater than or equal to the sum of their probabilities minus 1. This is known as Bonferroni's inequality.
02

Apply the formula for the probability of the union of two events

The probability of the union of two events E and F is given by: \( P(E 鈭� F) = P(E) + P(F) - P(E 鈭� F) \)
03

Rearrange the formula to solve for the intersection probability

We can rearrange the formula from Step 2 to solve for the probability of the intersection of events E and F: \( P(E 鈭� F) = P(E) + P(F) - P(E 鈭� F) \)
04

Apply the boundaries of the probabilities to conclude the inequality

Given that the probability of any event is between 0 and 1, we can conclude that the minimum value for P(E 鈭� F) equals 0, and the maximum value equals 1. Thus with the minimum value of 0, \( P(E 鈭� F) = P(E) + P(F) - 0 \) And with the maximum value of 1, \( P(E 鈭� F) \geq P(E) + P(F) - 1 \)
05

Verify with the given probabilities P(E) and P(F)

Now we can check the given probabilities to see if the inequality holds: \( P(E) = 0.9, P(F) = 0.8 \) According to the inequality from step 4: \( P(E 鈭� F) \geq 0.9 + 0.8 - 1 = 0.7 \) Thus, we have shown that the option \( P(E鈭〧) \geq 0.7 \) holds, and in general Bonferroni's inequality is true.

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!

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

A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from the box. What is the sample space? If, at all times, each marble in the box is equally likely to be selected, what is the probability of each point in the sample space?

Sixty percent of the families in a certain community own their own car, thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both?

Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box. What is the probability that the marble is black?

In a class there are four freshman boys, six freshman girls, and six sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?

What is the conditional probability that the first die is six given that the sum of the dice is seven?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.