EAS305
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