1 2 Outlines Probability Basic definitions: Randomization - - PowerPoint PPT Presentation

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Outlines

• Probability
• Basic definitions:

Randomization experiment Sample spaces Elementary outcomes

• Basic operations—conditional probability
• Bayes Theorem

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Randomness

 Things may happen randomly, for examples

• Comparison of treatment effects in clinical trials
• Calculation of the risk of breast cancer

 Probability

• Study of randomness
• Language of uncertainty

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Study of randomness

 Random Experiment : is the process of

• bserving the outcome of a chance event.

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Basic Definitions

 A random experiment for which the outcome

cannot be predicted with certainty

 But all possible outcomes can be identified

prior to its performance

 And it may be repeated under the same

conditions

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Basic Definitions

 Elementary outcomes: are all possible results

• f the random experiment.

 Sample space: is the set or collection of all the

elementary outcomes : {Head , Tail}

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Another ex.: Throw a single dice

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Another ex.: Throw a single die

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Sample space?

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Another ex.: Throw a pair of dice

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Sample space?

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Another ex.: Throw a pair of dice

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Sample space?

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Another ex.: Throw a pair of dice

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Sample space?

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Probability theory

 For a random experiment with n elementary outcomes

O1, O2, ... On, we assign a numerical weight or probability to each outcome.

 P(Oi )= the likelihood of the occurrence of an event  Toss a fair coin: P(H)=P(T)=0.5  Throw one die: P(die=5)=1/6  Throw two dice: P(Black 5, white 2)=1/36

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Properties of probability

Let Ω be the sample space for a probability measure P

• 0 ≤ P(Oi) ≤ 1, probability is number between 0

and 1; it is non-negative

• P(O1)+ P(O2)+...+P(On)= P(Ω) = 1
• A null set, denoted as ∅, has no elements and P(∅)

= 0

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Basic Operations

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An Event is a set of elementary outcomes. The probability of an event is the sum of the probabilities of the elementary outcomes in the set. For the dice example

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Event

Event description Event’s elementary outcomes probability A: dice add to 3 B:dice add to 6 C:White die shows 1 D: Black die shows 1

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An Event is a set of elementary outcomes. The probability of an event is the sum of the probabilities of the elementary outcomes in the set. For the dice example:

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Event

Event description Event’s elementary outcomes probability A: dice add to 3 {(1,2), (2,1)} B:dice add to 6 {(1,5),(2,4),(3,3),(4,2),(5,1)} C:White die shows 1 {(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)} D: Black die shows 1 {(1,1),(2,1),(3,1),(4,1),(5,1),(6,1)}

2 ( ) 36 P A = 5 ( ) 36 P B = 6 ( ) 36 P C = 6 ( ) 36 P D = 16

 Let Ω denote the set comprised of the totality of all

elements in our space of interest

• If A ⊂ Ω , Ā (complement of A) is the set of all

elements of which do not belong to A

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Basic definition

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Ω

A Ā

 For two events A and B,

• A or B( A ∪ B) : The event A or the event B occurs

(or both do)

• A and B (A ∩ B) : The event A and the event B both
• ccur
• Not A (Ā) : The event A does not occur

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Combining events using logical

• perations

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 Let A = {1, 2, 3} and B = {3, 4, 5}

• A ∩ B =
• A ∪ B =

 Let Ω = {1, 2, 3, 4, 5, 6, 7, 8, ...}: the positive integers, and let

A = {2, 4, 6, 8, . . .}

• Ā =

 A = {1, 2, 3} and B = {1, 2, 3, 4}

• Does A ⊂ B?

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Example

 Let C is the event White die is1 and D is the event Black die is

1 P(C)= P(D)= P(C ∩D)= P(C ∪D)=

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Example

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 Let C is the event White die is1 and D is the event Black die is

1 P(C)= P(D)= P(C ∩D)= P(C ∪D)=

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Example

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 P(C ∪D)= P(C) + P(D)- P(C ∩D)

P(C)=1/6 P(D)=1/6 P(C ∩D)= 1/36 P(C ∪D)=11/36

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Addition Rule

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 If the overlap of event A and B is empty, i.e. P(A ∩B)= 0 or A

∩B = Ø , in this case, we say event A and B are mutually exclusive. A: the dice add to 3 B: the dice add to 6

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Mutually exclusive

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 If A and B are mutually exclusive, then  P(A ∪B)= P(A) + P(B)

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Special Addition Rule

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Subtraction Rule

 P(Ā) = 1 – P(A)  A: a double 1 is not thrown  P(A)= ?

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Subtraction Rule

 P(Ā) = 1 – P(A)  A: a double-1 is not thrown  P(Ā)=1/36

So P(A)=1- 1/36 = 35/36

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Conditional Probability

Questions:

• 1. Throw a pair of dice. What’s the probability that the faces

sum to 3 (event A)? P(A)=?

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Conditional Probability

Questions:

• 1. Sample space of A

has 36 outcomes. P(A)=

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Conditional Probability

Questions:

• 1. Sample space of A

has 36 outcomes. P(A)=2/36=1/18

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Conditional Probability

Questions:

• 1. Throw a pair of dice. What’s the probability that the faces

sum to 3 (event A)? P(A)= 2/36=1/18

• 2. Suppose we throw the white die before the black die. The

white die comes up 1(event C). What’s the probability of A now? P(A|C)=? : Conditional Probability that event A will occur given event C has already occurred.

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Conditional Probability

Given event C is happened, the sample space is reduced. Reduced sample space of C has 6 outcomes.

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Conditional Probability

Given event C is happened, the sample space is reduced. Reduced sample space of C has 6 outcomes. A: the faces sum to 3 P(A|C)=

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Event A∩C

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Conditional Probability

Given event C is happened, the sample space is reduced. Reduced sample space of C has 6 outcomes. A: the faces sum to 3 P(A|C)=1/6

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Event A∩C

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Conditional Probability

A: the faces sum to 3 P(A|C)=1/6

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Event A∩C P(A∩C)=1/36 ( ) ( | ) ( ) P A C P A C P C ∩ =

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Event C, P(C)=1/6

Conditional Probability

The Conditional Probability of E given F is Fact 1: Fact 2: When E and F are mutually exclusive

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( ) ( | ) ( ) P E F P E F P F =  ( | ) 1 P E E = ( | ) P E F =

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Multiplication Rule

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( ) ( | ) ( ) ( | ) ( ) P E F P E F P F P F E P E = = 

Independence

When

When P(E)=P(E|F) or equivalently P(F)=P(F|E), E and F are independent When E and F are independent

Special Multiplication Rule

( ) ( ) ( ) P E F P E P F = 

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When

Event E and F are independent

( ) ( ) ( ) P E F P E P F =  ( | ) ( ) P E F P E = ( | ) ( ) P F E P F =

EX.

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Event C is white die comes up 1; event D is Black die comes up 1. Please show C and D are independent

2.

Event F is the sum of the two dice is 3. Show C and F are not independent

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 Addition Rule  Special Addition Rule: when E and F are mutually

exclusive

 Subtraction Rule  Multiplication Rule  Special Multiplication rule: when E and F are indep.

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Summary

( ) ( | ) ( ) ( | ) ( ) P E F P E F P F P F E P E = = 

P(Ā) = 1 – P(A)

( ) ( ) ( ) P E F P E P F = 

P(C ∪D)= P(C) + P(D)- P(C ∩D) P(C ∪D)= P(C) + P(D)

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Suppose a rare disease affects one out of every 1000 people in a population. And suppose that there is a good but not perfect test for this disease

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If a person has the disease, the test comes back positive 99% of the time

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About 2 % of uninfected patients also test positive Q: If your just tested positive, what are your chances of having the disease?

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Case of the false positive

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A: Patient has the disease B: Patient tests positive P(A)= P(B|A)=

P(B|Not A)= Q: P(A|B)=

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Case of the false positive

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A: Patient has the disease B: Patient tests positive P(A)=0.001 P(B|A)=0.99

P(B|Not A)=0.02 Q: P(A|B)=?

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Case of the false positive

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A: Patient has the disease B: Patient tests positive P(A)=0.001 P(B|A)=0.99

P(B|Not A)=0.02 Q: by conditional probability

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Case of the false positive

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( ) ( | ) ( ) P A B P A B P B ∩ =

A: Patient has the disease B: Patient tests positive P(A)=0.001 P(B|A)=0.99

P(B|Not A)=0.02 Q: By multiplication rule

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Case of the false positive

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( ) ( | ) ( ) ( | ) ( ) ( ) P A B P B A P A P A B P B P B ∩ = =

A: Patient has the disease B: Patient tests positive P(A)=0.001 P(B|A)=0.99

P(B|Not A)=0.02 Q: ??

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Case of the false positive

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( ) ( | ) ( ) ( | ) ( ) ( ) P A B P B A P A P A B P B P B ∩ = =

Total probability rule

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A: Disease status B: Test results The sample space can be divided into four mutually exclusive events

A not A B B ∩A B ∩ not A not B not B∩A not B ∩ not A

( ) ( ) ( ) ( | ) ( ) ( | ) ( ) P B P B A P B A P B A P A P B A P A = + = +  

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Bayes Theorem

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( ) ( | ) ( | ) ( | ) ( ) ( | ) ( ) P A P B A P A B P B A P A P B A P A = +

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A: Patient has the disease B: Patient tests positive P(A)=0.001 P(B|A)=0.99

P(B|Not A)=0.02 Q:

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Case of the false positive

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( ) ( | ) ( | ) ( | ) ( ) ( | ) ( ) 0.001*0.99 0.00099 0.99*0.001 0.02*0.999 0.02097 0.0472 P A P B A P A B P B A P A P B A P A = + = = + =

One exercise

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A1 A2

Ω

 For two events A1 and A2, please shade the area for

• A1 ∩ Ā2