Chapter 2: Q. 2.16 (page 53)

Use induction to generalize Bonferroni’s inequality to $n$ events. That is, show that
$P (E_{1} E_{2} \cdot \cdot \cdot E_{n}) \geq P (E_{1}) + \cdot \cdot \cdot + P (E_{n}) - (n - 1)$ .

Short Answer

Expert verified

proven by the principle of mathematical induction.

Step by step solution

Given Information.

Given, generalize Bonferroni’s inequality to $n$ events.

Explanation.

For events $E_{1}, E_{2}, \dots E_{n}$ .

$P (E_{1} E_{2} E_{3} \cdot \dots \cdot E_{n}) = P (E_{1}) + P (E_{2}) + \dots + P (E_{n}) - (n - 1)$

Proof by mathematical induction

For $n = 2$

$P (E_{1} E_{2}) = P (E_{1}) + P (E_{2}) - 1$

For proof of this refer to the Theoretical exerciserole="math" localid="1649259597999" $11$ .

If this is true for some $n \in ℕ$ . i.e.

$P (E_{1} E_{2} E_{3} \cdot \dots \cdot E_{n}) = P (E_{1}) + P (E_{2}) + \dots + P (E_{n}) - (n - 1)$

For events $E_{1}, E_{2}, \dots E_{n}, E_{n + 1}$

$P (E_{1} E_{2} E_{3} \cdot \dots \cdot E_{n} E_{n + 1}) = P [(E_{1} E_{2} E_{3} \cdot \dots \cdot E_{n}) E_{n + 1}]$

$= P (E_{1} E_{2} E_{3} \cdot \dots \cdot E_{n}) + P (E_{n + 1}) - 1$

$\overset{(1)}{=} P (E_{1}) + P (E_{2}) + \dots + P (E_{n}) - (n - 1) + P (E_{n + 1}) - 1 \overset{(2)}{=} P (E_{1}) + P (E_{2}) + \dots + P (E_{n}) + P (E_{n + 1}) - (n + 1 - 1)$

The statement holds for $n + 1$ thus by the principle of mathematical induction it holds for every $n \in ℕ$ .

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Use induction to generalize Bonferroni’s inequality to $n$ events. That is, show that
$P (E_{1} E_{2} \cdot \cdot \cdot E_{n}) \geq P (E_{1}) + \cdot \cdot \cdot + P (E_{n}) - (n - 1)$ .

Short Answer

Step by step solution

Given Information.

Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Use induction to generalize Bonferroni’s inequality to nevents. That is, show thatP(E1E2···En)≥P(E1)+···+P(En)−(n−1).

Short Answer

Step by step solution

Given Information.

Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Use induction to generalize Bonferroni’s inequality to $n$ events. That is, show that
$P (E_{1} E_{2} \cdot \cdot \cdot E_{n}) \geq P (E_{1}) + \cdot \cdot \cdot + P (E_{n}) - (n - 1)$ .