Conditional Probability

Bayes’ Theorem

Fall 2014 EAS 305 Lecture Notes

Prof. Jun Zhuang

University at Buﬀalo, State University of New York

September 10, ... 2014

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 1 of 26

Conditional Probability

Bayes’ Theorem

Agenda

1

Conditional Probability

Deﬁnition and Properties

Independence

General Deﬁnition

2

Bayes’ Theorem

Partition

Theorem

Examples

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 2 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Example

Example: Die. A = {2, 4, 6}, B = {1, 2, 3, 4, 5}. So Pr(A) = 1/2, Pr(B) = 5/6.

Suppose we know that B occurs. Then the prob of A “given” B is Pr(A|B) =

|A ∩ B|

2

=

5

|B|

So the prob of A depends on the info that you have! The info that B occurs allows us to regard B as a new, restricted sample space. And. . .

Pr(A|B) =

|A ∩ B|

|A ∩ B|/|S|

Pr(A ∩ B)

=

=

.

|B|

|B|/|S|

Pr(B)

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 3 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Deﬁnition: If Pr(B) > 0, the conditional prob of A given B is Pr(A|B) ≡ Pr(A ∩ B)/Pr(B).

Remarks: If A and B are disjoint, then Pr(A|B) = 0. (If B

occurs, there’s no chance that A can also occur.)

What happens if Pr(B) = 0? Don’t worry! In this case, makes no sense to consider Pr(A|B).

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 4 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Example: Toss 2 dice and take the sum.

A: odd toss = {3, 5, 7, 9, 11}

B: {2, 3}

Pr(A) = Pr(3) + · · · + Pr(11) =

4

2

1

2

+

+ ··· +

= .

36 36

36

2

1

2

1

+

=

.

36 36

12

Pr(A ∩ B)

Pr(3)

2/36

Pr(A|B) =

=

=

= 2/3.

Pr(B)

Pr(B)

1/12

Pr(B) =

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture...

Bayes’ Theorem

Fall 2014 EAS 305 Lecture Notes

Prof. Jun Zhuang

University at Buﬀalo, State University of New York

September 10, ... 2014

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 1 of 26

Conditional Probability

Bayes’ Theorem

Agenda

1

Conditional Probability

Deﬁnition and Properties

Independence

General Deﬁnition

2

Bayes’ Theorem

Partition

Theorem

Examples

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 2 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Example

Example: Die. A = {2, 4, 6}, B = {1, 2, 3, 4, 5}. So Pr(A) = 1/2, Pr(B) = 5/6.

Suppose we know that B occurs. Then the prob of A “given” B is Pr(A|B) =

|A ∩ B|

2

=

5

|B|

So the prob of A depends on the info that you have! The info that B occurs allows us to regard B as a new, restricted sample space. And. . .

Pr(A|B) =

|A ∩ B|

|A ∩ B|/|S|

Pr(A ∩ B)

=

=

.

|B|

|B|/|S|

Pr(B)

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 3 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Deﬁnition: If Pr(B) > 0, the conditional prob of A given B is Pr(A|B) ≡ Pr(A ∩ B)/Pr(B).

Remarks: If A and B are disjoint, then Pr(A|B) = 0. (If B

occurs, there’s no chance that A can also occur.)

What happens if Pr(B) = 0? Don’t worry! In this case, makes no sense to consider Pr(A|B).

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 4 of 26

Conditional Probability

Bayes’ Theorem

Deﬁnition and Properties

Independence

General Deﬁnition

Example: Toss 2 dice and take the sum.

A: odd toss = {3, 5, 7, 9, 11}

B: {2, 3}

Pr(A) = Pr(3) + · · · + Pr(11) =

4

2

1

2

+

+ ··· +

= .

36 36

36

2

1

2

1

+

=

.

36 36

12

Pr(A ∩ B)

Pr(3)

2/36

Pr(A|B) =

=

=

= 2/3.

Pr(B)

Pr(B)

1/12

Pr(B) =

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture...

Conditional Probability

Bayes’ Theorem

Fall 2014 EAS 305 Lecture Notes

Prof. Jun Zhuang

University at Buﬀalo, State University of New York

September 10, ... 2014

Prof. Jun Zhuang Fall 2014 EAS 305 Lecture Notes Page 1 of 26

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