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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鈭�E2F=PE1鈭�F鈬�PE1E2鈭�F=PE1鈭�FPE2鈭�F

Conditional independence is independence in conditional probability.

Generalize the formula for independence of multiple events:

nevents E1,E2,鈥�,Enare conditionally independent given Fif

for any kand any different j1,j2,鈥�,jk鈭�{1,2,鈥�,n}

P鈭�i=1kEji鈭�F=鈭�i=1kPEji鈭�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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In Example 3f, suppose that the new evidence is subject to different possible interpretations and in fact shows only that it is 90 percent likely that the criminal possesses the characteristic in question. In this case, how likely would it be that the suspect is guilty (assuming, as before, that he has the characteristic)?

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On rainy days, Joe is late to work with probability .3; on nonrainy days, he is late with probability .1. With probability .7, it will rain tomorrow.

(a) Find the probability that Joe is early tomorrow.

(b) Given that Joe was early, what is the conditional probability that it rained?

In any given year, a male automobile policyholder will make a claim with probability pmand a female policyholder will make a claim with probability localid="1646823185045" pf,where pf鈮�pm. The fraction of the policyholders that are male is 伪,0<伪<1.A policyholder is randomly chosen. If Aidenotes the event that this policyholder will make a claim in the year i,show that

PA2鈭�A1>PA1

Give an intuitive explanation of why the preceding inequality is true.
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