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A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 2: Q. 2.16 (page 53)

Use induction to generalize Bonferroni鈥檚 inequality to $n$ events. That is, show that
$P (E_{1} E_{2} 路路路 E_{n}) 鈮� P (E_{1}) + 路路路 + P (E_{n}) 鈭� (n 鈭� 1)$ .

Short Answer

Expert verified

proven by the principle of mathematical induction.

Step by step solution

01

Given Information.

Given, generalize Bonferroni鈥檚 inequality to $n$ events.

02

Explanation.

For events $E_{1}, E_{2}, 鈥� E_{n}$ .

$P (E_{1} E_{2} E_{3} 路鈥� 路 E_{n}) = P (E_{1}) + P (E_{2}) + 鈥� + P (E_{n}) - (n - 1)$

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 $n 鈭� 鈩�$ . i.e.

$P (E_{1} E_{2} E_{3} 路鈥� 路 E_{n}) = P (E_{1}) + P (E_{2}) + 鈥� + P (E_{n}) - (n - 1)$

For events $E_{1}, E_{2}, 鈥� E_{n}, E_{n + 1}$

$P (E_{1} E_{2} E_{3} 路鈥� 路 E_{n} E_{n + 1}) = P [(E_{1} E_{2} E_{3} 路鈥� 路 E_{n}) E_{n + 1}]$

$= P (E_{1} E_{2} E_{3} 路鈥� 路 E_{n}) + P (E_{n + 1}) - 1$

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

The statement holds for $n + 1$ thus by the principle of mathematical induction it holds for every $n 鈭� 鈩�$ .

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Most popular questions from this chapter

Five balls are randomly chosen, without replacement, from an urn that contains $5$ red, $6$ white, and $7$ blue balls. Find the probability that at least one ball of each color is chosen.

A deck of cards is dealt out. What is the probability that the $14$ th card dealt is an ace? What is the probability that the first ace occurs on the $14$ th card?

Two cards are randomly selected from an ordinary playing deck. What is the probability that they form a blackjack? That is, what is the probability that one of the cards is an ace and the other one is either a ten, a jack, a

queen, or a king?

Five people, designated as $A, B, C, D, E$ , are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that

(a) there is exactly one person between $A$ and $B$ ?

(b) there are exactly two people between $A$ and $B$ ?

(c) there are three people between $A$ and $B$ ?

${鈭�}_{1}^{鈭�} E_{i} F = ({鈭�}_{1}^{鈭�} E_{i}) F$ and ${鈭�}_{1}^{鈭�} (E_{i} 鈭� F) = ({鈭�}_{1}^{鈭�} E_{i}) 鈭� F$

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