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Chapter 3. Conditional Probability and Independence

Section 3.3. Theorem of Total Probability and Bayes’ Rule

Section 3.4 Odds, Odds Ratios, and Relative Risk

Jiaping Wang

Department of Mathematical Science

01/30/2013, Wednesday

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Outline

Theorem of Total Probability

Bayes’ Rule Odds, Odds Ratios and Relative Risk

Homework #3

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Part 1. Theorem of Total Probability

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Theorem 3.2

Consider an example, if there is a partition B1 and B2 such that B1UB2=S and B1∩B2=ø, then we can find A = (A∩B1) U (A∩B2) and thus P(A)=P(A∩B1) +P(A∩B2)=P(A|B1)P(B1)+P(A|B2)P(B2)

Theorem of Total Probability:If B1, B2, …, Bk is a collection of mutually exclusive and

exhaustive events, then for any event A, we have

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A company buys microchips from three suppliers-I, II, and III. Supplier I has a record of providing microchips that contain 10% defectives; Supplier II has a defective rate of 5% and Supplier III has a defective rate of 2%. Suppose that 20%, 35% and 45% of the current supply came from Suppliers I, II, and III, respectively. If a microchip is selected at random from this supply, what is the probability that it is defective?

Example 3.8

Solution: BI={Chip comes from Supplier I}, BII, BIII, D denote defective, ND – non-defective. P(BI∩D)=0.20(0.10)=0.02, P(BI∩ND)=0.18, P(BII∩D)=0.175, P(BII∩ND)=0.3325, P(BIII∩D)=0.009, P(BIII∩ND)=0.441. P(BI)=0.20, P(BII)=0.35, P(BIII)=0.45. So by Law of Total Probability, P(D)=P(D|BI)P(BI)+P(D|BII)P(BII)+P(D|BIII)P(BIII)=0.0465.

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Part 2. Bayes’ Rule

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Theorem 3.3

Bayes’ Rule. If the events B1, B2, …, Bk form a partition of the sample space S, and A is any event in S, then

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Example 3.9

Consider again the information from Example 3.8. If a random selected microchip is defective, what is the probability that it came from Supplier II?

Solution: By Bayes’ rule, P(BII|D)=P(D|BII)P(BII)/P(D)=0.376

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Part 3. Odds, Odds Ratios, and Relative Risk

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An Example

The Physicians’ Health study on the effects of aspirin on heart attacks randomly assigned to 22,000 male physicians to either the “aspirin” or “placebo” arm of the study. The data on myocardial infarctions (MI) are given in table.

MI No MI TotalAspirin 139 10,898 11,037Placebo 239 10,795 11034Total 378 21,683 22,071

One my talk about what are the odds in favor of MI over non-MI. The odds in favor of anevent A is the ratio of the probability of A to the probability of the complement of A. For aspirin P(MI)/P(non-MI)=(139/11037)/(10898/11037)=139/10898=0.013; For placebo,P(MI)/P(non-MI)=239/10796=0.022, which shows odds of heart attach with placebo ishigher than the risk with aspirin,

the odds ratio = Odds of MI with aspirin/odds of MI without aspirin=0.59<1.

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Definitions of Odds, Odds Ratios, andRelative Risk

Yes NoA a bB c d

Odds ratios form a useful summary of the frequencies in a 2x2(two-way) frequency table.

The odds in favor of A =a/b, odds in favor of B=c/dThe odds ratio = a*d/b*c.The relative risk is the ratio of the probability of an event in the treatment group to the probability of same event in the placebo group,Relative risk = P(Yes|A)/P(Yes|B)=[a/(a+b)]/[c/(c+d)]=a(c+d)/c(a+b).

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Example 3.10

The Physicaians’ Health Study included only men and the results clearly indicated that taking a low dose of aspirin reduced the risk of MI. In 2005, the results of the Women’s Health study were published in the table. This study randomized almost 40,000 women, ages 45 and older, to either aspirin or placebo and followed the women for 10 years.

MI No MI TotalAspirin 198 19,736 19,934Placebo 193 19,749 19,942Total 391 39,485 39,876

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Example 3.10 Continue

a. Find the odds of MI for the aspirin group.b. Find the odds of MI for the placebo group.c. Find the odds ratio of MI for the aspirin and placebo

groups.d. Find the relative risk of MI for the aspirin and placebo

group.

Solutions: a. Odds for the aspirin: P(MI)/P(non-MI)=198/19736 b. Odds for the placebo: P(MI)/P(non-MI)=193/19749 c. Odds ratio = 198/19736*19749/193=1.01 > 1 d. Relative risk = 198/19934*19942/193=1.03>1

Which means low-dose aspirin regime is not effective for reducing MI for women.

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Homework 3

Page 66: 3.2Page 67: 3.6, 3.11, 3.14Page 76: 3.27Page 77: 3.32, 3.34, 3.36Page 81: 3.44Page 86: 3.55

Due on Wednesday, 02/06/2013.