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Chapter 2: Q. 2.14 (page 55)

Prove Boole’s inequality:

P∪i=1∞Ai≤∑i=1∞PAi

Short Answer

Expert verified

Proof by mathematical induction:

Assume that equality stands for somen, and it follows that inequality stands forn+1.

Step by step solution

01

Given Information.

Prove that for all eventsA1,A2,...An.

P∪k=1nAk≤∑k=1nPAk

02

Explanation.

Proof by mathematical induction:

Forn=1the statement is true

PA1≤PA1

Hypothesis:P∪k=1nAk≤∑k=1nPAkthis is true for anynevent.

Then for any n+1events:

P∪k=1n+1Ak=P∪k=1nAk∪An+1associative law

&=P∪k=1nAk+PAn+1-P∪k=1nAk∩An+1≤∑k=1nPAk+PAn+1-P∪k=1nAk∩An+1≤∑k=1n+1PAk-P∪k=1nAk∩An+1≤∑k=1n+1PAk

P∪n+1Ak≤∑k=1n+1PAk

This proves the statement, by the principle of mathematical induction.

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Most popular questions from this chapter

A pair of dice is rolled until a sum of either 5or 7appears. Find the probability that a 5occurs first.

Hint: Let Endenote the event that a 5occurs on the nth roll and no 5or7occurs on the first n-1rolls. Compute P(En) and argue that ∑n=1∞P(En)is the desired probability

If two dice are rolled, what is the probability that the sum of the upturned faces equalsi? Find it fori=2,3,...,11,12.

Consider an experiment that consists of determining the type of job—either blue collar or white collar— and the political affiliation—Republican, Democratic, or Independent—of the 15 members of an adult soccer team.

How many outcomes are

(a) in the sample space?

(b) in the event that at least one of the team members is a blue-collar worker?

(c) in the event that none of the team members considers himself or herself an Independent?

The chess clubs of two schools consist of, respectively, 8and9 players. Four members from each club are randomly chosen to participate in a contest between the two schools. The chosen players from one team are then randomly paired with those from the other team, and each pairing plays a game of chess. Suppose that Rebecca and her sister Elise are on the chess clubs at different schools. What is the probability that

(a) Rebecca and Elise will be paired?

(b) Rebecca and Elise will be chosen to represent their schools but will not play each other?

(c) either Rebecca or Elise will be chosen to represent her school?

An urn contains 5red, 6blue, and 8green balls. If a set of 3balls is randomly selected, what is the probability that each of the balls will be

(a) of the same color?

(b) of different colors? Repeat under the assumption that whenever a ball is selected, its color is noted and it is then replaced in the urn before the next selection. This is known as sampling with replacement .
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