Transcript The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL A

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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Example 3.8
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?
Solution: BI={Chip comes from Supplier I}, BII, BIII, D denote defective, ND – nondefective. 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
Total
Aspirin
139
10,898
11,037
Placebo
239
10,795
11034
Total
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 an
event 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 is
higher 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, and
Relative Risk
Odds ratios form a useful summary of the frequencies in a
2x2(two-way) frequency table.
Yes
No
A
a
b
B
c
d
The odds in favor of A =a/b, odds in favor of B=c/d
The 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
Total
Aspirin
198
19,736
19,934
Placebo
193
19,749
19,942
Total
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: 3.2
Page 67: 3.6, 3.11, 3.14
Page 76: 3.27
Page 77: 3.32, 3.34, 3.36
Page 81: 3.44
Page 86: 3.55
Due on Wednesday, 02/06/2013.
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