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Transcript 
COMP 170 L2
L16: Conditional Probability and Independence
Page 1
COMP 170 L2
Page 2
Outline



Conditional probability

Motivating examples

Definition
Independence

Definition

Property

Examples
Independent trials processes
COMP 170 L2
Example: Roll Two Dice
COMP 170 L2
Event F and Change of Sample Spaces
COMP 170 L2

F favors none of the three remaining possible outcomes

New probabilities

Preserve ratios

Sum to one
COMP 170 L2
COMP 170 L2
COMP 170 L2
COMP 170 L2
COMP 170 L2
Page 10
Outline


Conditional probability

Motivating examples

Definition
Independence

Definition

Property


Examples
Independent trials processes
COMP 170 L2
COMP 170 L2
COMP 170 L2
COMP 170 L2
A Note
COMP 170 L2
Page 15
Use of Conditional Probability
COMP 170 L2
COMP 170 L2
Page 17
Outline



Conditional probability

Motivating examples

Definition
Independence

Definition

Property

Examples
Independent trials processes
COMP 170 L2
Page 18
Motivating Example

Learning that F occurred does not change probability of E

F is irrelevant to E

E is independent of F
COMP 170 L2
Page 19
Definition
COMP 170 L2
Page 20
Definition
COMP 170 L2
Page 21
Outline



Conditional probability

Motivating examples

Definition
Independence

Definition

Property

Examples
Independent trials processes
COMP 170 L2
Page 22
COMP 170 L2
Page 23
COMP 170 L2
Page 24
Outline



Conditional probability

Motivating examples

Definition
Independence

Definition

Property

Examples
Independent trials processes
COMP 170 L2
Page 25
COMP 170 L2
Page 26
COMP 170 L2
Page 27
COMP 170 L2
Page 28
COMP 170 L2
Page 29
Outline



Conditional probability

Motivating examples

Definition
Independence

Definition

Property

Examples
Independent trials processes
COMP 170 L2
Page 30
Multiple Stage Process




Recall

Probability associated with a process with uncertain outcomes

Sample space: set of all possible outcomes

Event: subset of sample space

Probability distribution: weight for each element
One stage process

Flipping a coin once

Throwing two dice (all at once)
Multiple stage process

Flip coin multiple times

Hashing: Hash keys one at a time
Next: Discuss probabilities associated with multiple stage processes
COMP 170 L2
Page 31
Independent Trial Process
Outcome at stage i independent of outcomes at earlier stages
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COMP 170 L2
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COMP 170 L2
Page 39
Recap: 04-05-2010

Probability starts with a process (test, experiment) whose outcome is
uncertain

Sample space: the set of all possible outcomes


Might not be unique
Consider the probability of an event E

At the beginning, we have some assessment: P(E)

Then we get a new piece of information:
 “Event F occurred”

Afterwards, we talk about the probability of E given F
COMP 170 L2
Recap: 04-05-2010

If P(E|F) = P(E)

The information that “Event F occurred” does not change the
probability of E

So, E is independent of F
COMP 170 L2
Recap: 06-05-2010
Outcome at stage i independent of outcomes at earlier stages
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