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Chapter 3: Q. 3.1 (page 106)

Show that if P(A)>0, then

P(AB∣A)≥P(AB∣A∪B)

Short Answer

Expert verified

We proved that P(AB∣A)≥P(AB∣A∪B)by applying conditional probability asP(A)>0.

Step by step solution

01

Given Information

IfP(A)>0, We have to prove thatP(AB∣A)≥P(AB∣A∪B).

02

Explanation

AB⊆(A∪B)⇒AB∩(A∪B)=AB

The conditional probability P[AB∣(A∪B)]can be reduced to

P[AB∣(A∪B)]=P[AB∩(A∪B)]P(A∪B)=P(AB)P(A∪B)

Again, from set arithmetic

A⊆A∪B⇒P(A)≤P(A∪B)

Finally

P(AB∣A)=P(AB)P(A)≥P(AB)P(A∪B)=P[AB∣(A∪B)]

03

Final Answer

P(AB∣A)- is the percentage of Athat is in role="math" localid="1647853157806" AB

P[AB∣(A∪B)]- is the percentage of A∪Bthat is in AB.

And since A∪B is larger,P[AB∣(A∪B)]≤P(AB∣A).

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Most popular questions from this chapter

An urn contains 6white and9black balls. If 4balls are to be randomly selected without replacement, what is the probability that the first 2selected is white and the last 2 black?

A high school student is anxiously waiting to receive mail telling her whether she has been accepted to a certain college. She estimates that the conditional probabilities of receiving notification on each day of next week, given that she is accepted and that she is rejected, are as follows:

DayP(mail/accepted)P(mail/rejected)
Monday.15
.05
Tuesday.20
.10
Wednesday.25
.10
Thursday.15
.15
Friday.10
.20

She estimates that her probability of being accepted is .6.

(a) What is the probability that she receives mail on Monday?

(b) What is the conditional probability that she receives mail on Tuesday given that she does not receive mail on Monday?

(c) If there is no mail through Wednesday, what is the conditional probability that she will be accepted?

(d) What is the conditional probability that she will be accepted if mail comes on Thursday?

(e) What is the conditional probability that she will be accepted if no mail arrives that week?

A ball is in any one of nboxes and is in the ith box with probability Pi. If the ball is in box i, a search of that box will uncover it with probability αi. Show that the conditional probability that the ball is in box j, given that a search of box idid not uncover it, is

Pj1-αiPiifj≠i

1-αiPi1-αiPiifj=i

In a class, there are 4 first-year boys, 6 first-year girls, and 6 sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at random?

In Example 5e, what is the conditional probability that the ith coin was selected given that the first n trials all result in heads?
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