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Chapter 3: Q.3.27 (page 109)

Extend the definition of conditional independence to more than 2 events.

Short Answer

Expert verified

Conditional independence is independence in conditional probability.

Step by step solution

01

Given Information 

The definition of conditional independence to more than 2 events.

02

Explanation

Conditional independence of two events -E1,E2, with condition Fis defined by any of two equivalent conditions:

PE1∣E2F=PE1∣F⇔PE1E2∣F=PE1∣FPE2∣F

Conditional independence is independence in conditional probability.

Generalize the formula for independence of multiple events:

nevents E1,E2,…,Enare conditionally independent given Fif

for any kand any different j1,j2,…,jk∈{1,2,…,n}

P∩i=1kEji∣F=∏i=1kPEji∣F

03

Final Answer

Multiple events are conditionally independent if conditional probability of intersection of any subset of events is the product of the conditional probability of those events.

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Most popular questions from this chapter

An urn has r red and w white balls that are randomly removed one at a time. Let Ribe the event that the ith ball removed is red. Find

a). P(Ri)

b). PR5∣R3

c).PR3∣R5

Consider an urn containing 12balls, which 8are white. A sample of size4is to be drawn with replacement (without replacement). What is the conditional probability (in each case) that the first and third balls drawn will be white given that the sample drawn contains exactly3 white balls?

Repeat Problem 3.84 when each of the 3 players

selects from his own urn. That is, suppose that there are

3 different urns of 12 balls with 4 white balls in each urn.

Suppose that we want to generate the outcome of the flip of a fair coin, but that all we have at our disposal is a biased coin that lands on heads with some unknown probability p that need not be equal to 1 2 . Consider the following procedure for accomplishing our task: 1. Flip the coin. 2. Flip the coin again. 3. If both flips land on heads or both land on tails, return to step 1. 4. Let the result of the last flip be the result of the experiment.

(a) Show that the result is equally likely to be either heads or tails.

(b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?

A total of 48percent of the women and 37percent of the men who took a certain“quit smoking” class remained nonsmokers for at least one year after completing the class. These people then attended a success party at the end of the year. If 62percent of the original class was male,

(a) what percentage of those attending the party were women?

(b) what percentage of the original class attended the party?
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