Laws of Probability

43 Pages·2015·2.31 MB·English

by  Gittelson, Simone N.

NEAFS Probabilistic DNA Mixture 9/25/2015
Interpretation Workshop
NEAFS
Probabilistic DNA Mixture
Interpretation Workshop
Allentown, PA
September 25-26, 2015
Probability Theory and
Likelihood Ratios
Simone Gittelson, Ph.D., [email protected]
Michael Coble, Ph.D., [email protected]
Acknowledgement
I thank Michael Coble, Bruce Weir and John Buckleton
for their helpful discussions.
Disclaimer
Points of view in thispresentation are mine and do
not necessarilyrepresent the official position or
policies of the National Institute of Standards and
Technology.
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NEAFS Probabilistic DNA Mixture 9/25/2015
Interpretation Workshop
Probability Theory
Vocabulary and Notation
rolling a 6 and a 1
it will snow tomorrow
observing a peak for allele 8
8
Mr. X stabbed Mr. Y
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Vocabulary and Notation
EVENT PROPOSITION
𝑅
𝑆
𝐸
8
𝐻
Vocabulary and Notation But I do not know
which one is true?!?
I am uncertain.
COMPLEMENT event or proposition

𝑅 𝑅

𝑆 𝑆

𝐸 𝐸
8

𝐻 𝐻
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Vocabulary and Notation
PROBABILITY
Pr(𝑅)
Pr(𝑆)
Pr(𝐸)
8
Pr(𝐻)
Laws of Probability
Law #1: A probability can take any value between 0 and 1,
including 0 and 1.
certain
1
certaintythatstatementistrue
0.75
EXAMPLE: rollinga 6-sided die
0.66
𝑃𝑟 1,2,3,4,5𝑜𝑟6 =1
0.5 𝑃𝑟 7 =0
0.33
0.25
0 certaintythatstatementisfalse
impossible
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Laws of Probability
Law #2: The probability of event A orevent B occurring is
equal to the probability of event A plusthe probability of
event Bminus the probability of event A and B.
Pr 𝐴 𝑜𝑟 𝐵 = Pr 𝐴 + Pr 𝐵 − Pr(𝐴 𝑎𝑛𝑑 𝐵)
𝑃𝑟 𝐴𝑜𝑟𝐵
= 𝑃𝑟 𝐴 +𝑃𝑟 𝐵
event A event B
𝑃𝑟 𝐴𝑜𝑟𝐴
= 𝑃𝑟 𝐴 +𝑃𝑟 𝐴
= 1
A: rolling an odd number B: rolling a 2
Second Law of Probability
Blood groups:
O A B AB
0.45 0.40 0.11 0.04
1. What is the probability that a person drawn
randomly from this population is blood group type A
or AB?
2. What is the probability that a person drawn
randomly from this population is not blood group
type AB?
data from: http://www.redcrossblood.org/learn-about-blood/blood-types
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1. What is the probability that a person drawn randomly
from this population is blood group type A or AB?
Blood groups:
O A B AB
0.45 0.40 0.11 0.04
A. 0.016
B. 0.36
C. 0.44
D. 0.51
E. 10
0% 0% 0% 0% 0% 0%
0.016 0.36 0.44 0.51 0.55 10
2. What is the probability that a person drawn randomly
from this population is not blood group type AB?
Blood groups:
O A B AB
0.45 0.40 0.11 0.04
A. 0.04
B. 0.45
C. 0.51
D. 0.85
E. 0.96
0% 0% 0% 0% 0% 0%
0.04 0.45 0.51 0.56 0.85 0.96
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Laws of Probability
Law #2: The probability of event A orevent B occurring is
equal to the probability of event A plusthe probability of
event Bminus the probability of event A and B.
Pr 𝐴 𝑜𝑟 𝐵 = Pr 𝐴 + Pr 𝐵 − Pr(𝐴 𝑎𝑛𝑑 𝐵)
𝑃𝑟 𝐴𝑜𝑟𝐵 = 𝑃𝑟 𝐴 +𝑃𝑟 𝐵
−𝑃𝑟 𝐴𝑎𝑛𝑑 𝐵
event A event B
A: rolling an odd number B: rolling a number greater
than 2
Second Law of Probability
Blood groups:
O A B AB
0.45 0.40 0.11 0.04
3. What is the probability that a person drawn
randomly from this population has a blood group
that contains an A (groups A and AB) or a blood
group that contains a B (groups B and AB)?
data from: http://www.redcrossblood.org/learn-about-blood/blood-types
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3. What is the probability that a person drawn randomly from this
population has a blood group that contains an A (groups A and
AB) or a blood group that contains a B (groups B and AB)?
Blood groups:
O A B AB
0.45 0.40 0.11 0.04
A. 0.0204
B. 0.45
C. 0.55
D. 0.59
E. 14.75
0% 0% 0% 0% 0% 0%
0.0204 0.45 0.51 0.55 0.59 14.75
Laws of Probability
Law #3: The probability of event A andevent B occurring is
equal to the probability of event A timesthe probability of
event B.
Pr 𝐴 𝑎𝑛𝑑 𝐵 = Pr 𝐴 × Pr 𝐵|𝐴
= Pr(𝐵) × Pr(𝐴|𝐵)
event A event B
A: rolling an odd number B: rolling a 3 or less
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How Tall is Stefanie?
Conditional Probabilities
100
B
10
A
40
10
What is 𝑃𝑟 𝐵 ?
100
10
What is 𝑃𝑟 𝐵|𝐴 ?
40
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Third Law of Probability
(independent events)
Blood groups:
O A B AB
0.45 0.40 0.11 0.04
Rh factor:
+ -
0.82 0.18
4. What is the probability that a person drawn randomly
from this population is blood group type A and Rh+?
data from: http://www.redcrossblood.org/learn-about-blood/blood-types
4. What is the probability that a person drawn randomly
from this population is blood group type A and Rh+?
Blood groups: Rh factor:
O A B AB + -
0.45 0.40 0.11 0.04 0.82 0.18
A. 0.328
B. 0.4
C. 0.42
D. 0.82
E. 1.22
0% 0% 0% 0% 0% 0%
0.072 0.328 0.4 0.42 0.82 1.22
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