Q. 2.16

Question

Use induction to generalize Bonferroni’s inequality to $n$ events. That is, show that

$P (E_{1} E_{2} \cdot \cdot \cdot E_{n}) \geq P (E_{1}) + \cdot \cdot \cdot + P (E_{n}) - (n - 1)$ .

Step-by-Step Solution

Verified

Answer

proven by the principle of mathematical induction.

1Step 1 Given Information.

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

2Step 2 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 exercise $11$ .

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 $n \in ℕ$ .

Q. 2.15

Q. 2.17

Other exercises in this chapter

Q. 2.13

Prove thatP(EFc) = P(E) − P(EF).

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Q. 2.15

An urn contains M white and N black balls. If a random sample of size ris chosen, what is the probability that it contains exactly kwhite balls?

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Q. 2.17

Consider the matching problem, Example5m, and define it to be the number of ways in which theN men can select their hats so that no man selects his own.Arg

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Q. 2.18

Let fndenote the number of ways of tossing a coin n times such that successive heads never appear. Argue that fn = fn−1 + fnͨ

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