Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Chapter 2 - Lecture 4
Conditional Probability
Andreas Artemiou
September 21st, 2009
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Review
Conditional Probability
Definition
Notation
Formulas
Law of Total Probability
Bayes’ Rule
Exercises
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Probability
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Until now we have learn how to assign probabilities on sets
that are consisted of simple events without assuming that we
know anything regarding the environment that might affect
the probability.
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Conditional Probability
I
I
Conditional Probability is the probability than an event will
occur when we know some facts that have already happen.
Example:
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I have a class of 100 students
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What is the probability that student A has failed the class?
What is the probability that student A has failed the class
given that 15 students have failed the class?
What is the probability that student A has failed the class
given that all the students have failed the class?
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Notation
I
Conditional Probability of event A given that the event
B has occurred:
P(A|B)
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Example
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Let say that I have 100 CDs from each of the two brands
CDMaker and MakingCD
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Let say that 5 of the CDs from CDMaker are defective and 10
of the CDs from MakingCD are defective.
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Denote with A the event that a CD is defective and with B
the event that a CD is branded from CDMaker.
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Find P(A), P(B), P(Ac ), P(A|B), P(Ac |B), P(A|B c ),
P(B|A), P(B c |Ac ).
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Formula
I
In the previous example we constructed a table to find
Conditional probabilities. If we are not able to construct a
Table then we can use the following formula to find
conditional probabilities:
P(A|B) =
Andreas Artemiou
P(A ∩ B)
P(B)
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Example
I
Let say 50% of the CDs I own are CDMaker branded and
7.5% are defective and 2.5% are both branded CDMaker and
defective.
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Denote with A the event that a CD is defective and with B
the event that a CD is branded from CDMaker.
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What is the P(the CD is defective — the CD is branded
CDMaker) = P(A|B)?
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Formula
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The previous formula gave us another way of calculating the
Probability of events ”A and B, as follows:
P(A ∩ B) = P(A|B)P(B)
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Example 2.29 page 77
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Definition
Notation
Formulas
Example 2.29 page 77 (using tree diagrams)
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Law of Total Probability
I
Let A1 , . . . , Ak be mutually exclusive and exhaustive events.
Then for any event B,
P(B) =
k
X
P(B|Ai )P(Ai )
i=1
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The events A1 , . . . , Ak are said to be exhaustive if one of
them must occur, that is A1 ∪ . . . ∪ Ak = S
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Bayes’ Rule
I
Let A1 , . . . , Ak be mutually exclusive and exhaustive events
with P(Ai ) > 0, ∀i
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Then for any event B for which P(B) > 0
P(Aj |B) =
P(Aj ∩ B)
P(B|Aj )P(Aj )
= k
P(B)
X
P(B|Ai )P(Ai )
i=1
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Example
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Denote with A the event that a CD is defective and with B
the event that a CD is branded from CDMaker.
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We know that 7.5% of the CDs I own are defective
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If a CD is defective there is 33.33% probability that it is
branded CDMaker. If it is not defective there is 51.35% that
it was made by CDMaker
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Find P(A|B) and P(Ac |B) first by constructing a tree
diagram and then by using the formula.
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability
Outline
Review
Conditional Probability
Law of Total Probability
Bayes’ Rule
Exercises
Exercises
I
Section 2.4 page 80
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Exercises 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58,
59, 60, 61, 62, 63, 64, 65
Andreas Artemiou
Chapter 2 - Lecture 4 Conditional Probability