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Transcript
Today
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Today: Some more counting examples; Start Chapter 2
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Important Sections from Chapter 1: 1.1-1.5; Please read 1.7-1.8
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Reading:
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–
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Assignment #2 is up on the web site
www.stat.lsa.umich.edu/~dbingham/stat405
Please read Chapter 2
Suggested problems: 2.4, 2.5, 2.7, 2.13, 2.25, 2.28, 2.32, 2R1, 2R2
Example
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A batch of 50 manufactured items contains 5 defective items. If 10
items are randomly selected what is the probability that at least one
of the defectives is found?
Example
•
If there are 60 people in a Statistics 405 class, what is the probability
that at least 2 have the same birthday?
Partitions
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A partition of an event F is a collection of events (E1, E2, …, Ek) that
are mutually disjoint (i.e., EiEj= { }, for all i and j ) and where
F  E1  E2  ...  Ek
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Since none of the Ei’s have outcomes in common, then the number of
outcomes in F is the same as the sum of the outcomes in each of the
Ei’s
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That is, if F has N outcomes and Ei has Ni outcomes, then
N=N1+N2…+Nk
Partitions
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If E1, E2,…, Ek form a partition of F, then the number of
distinguishable ways to partition N objects into k groups is
N


N!

 
 N1 , N 2 ,... N k  N1!N 2 !...  N k !
Example
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What is the number of ways that 52 cards may be divided equally
among 4 players
Chapter 2
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Chapter 2 has two main topics:
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–
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Random variables and their distributions
Conditional probability and independence
Will begin with the second topic
Conditional Probability
Example:
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A family has two children
The sample space for the possible gender of their children is {gg, gb, bg,
bb}.
The parents are seen leaving a Girl Scout meeting, suggesting that at
least one of their children is a girl
If each gender is equally likely, what is the probability that the other
child is a boy
Conditional Probability
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•
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Consider a probability model with sample space 
Let E be an event in the sample space
If F is another event in the sample space where P(F)>0, the
conditional probability of E given F is:
P( E | F ) 
•
P( EF )
P( F )
Idea: This represents the appropriate change in the probability
assignment when we know that F has occurred
Example
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From a group of 5 Democrats, 5 Republicans and 5 Independents, a
committee of size 3 is to selected
•
What is the probability that each group will be represented on the
committee if the first person selected is an Independent?
Some Useful Formulas
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Multiplication Rule:
P( EF )  P( E ) P( F | E )  P( F ) P( E | F )
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Law of Total Probability:
P ( E )  P( E | F ) P ( F )  P ( E | F c ) P( F c )
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Bayes Theorem:
P( F | E ) 
P( E | F ) P( F )
P( E | F ) P( F )  P( E | F c ) P( F c )
Where do these come from?
Example
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•
•
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Consider a routine diagnostic test for a rare disease
Suppose that 0.1% of the population has the disease, and that when
the disease is present the probability that the test indicates the disease
is present is 0.99
Further suppose that when the disease is not present, the probability
that the test indicates the disease is present is 0.10
For the people who test positive, what is the probability they actually
have the disease
Several Events
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Suppose (E1, E2, …, Ek) form a partition of the sample space
k
P( F )   P( F | Ei ) P( Ei )
i 1
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Bayes Theorem: suppose (E1, E2, …, Ek) form a partition of the
sample space
P( E j | F ) 
P( F | E j ) P( E j )
k
 P ( F | E i ) P ( Ei )
i 1
Example (2.28 from text)
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•
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Evidence in a paternity suit indicates that 4 particular men are
equally likely to be the father of the child
Both the child and the mother have type O blood
The table below give the blood type and the probability of producing
a type O offspring with a type O mother
What is the probability that man 1 is the father? Man 3? Man 3?
Man
1
2
3
4
Blood Type
A
B
O
AB
P(O child|man and O-type mother)
.431
472
1
0
Independence
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For a given probability model and events E and F, the two events are
said to be independent iff P(EF)=P(E)P(F)
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Another way of viewing this is that for events E and F, the two
events are said to be independent iff P(E|F)=P(E)
Example
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A single card is drawn from a standard deck
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A={Ace}
B={Spade}
Are these events independent?
Example
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Consider a pair of genes which may be one of two alleles a or A
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This could represent a smooth (a) or wrinkled seed (A)
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According to Medelain Theory, each parent contributes either a or A
to the offspring independently
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E1={parent 1 gives a}
E2={parent 2 gives a}
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P(aa)=