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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 Notes

Page 5 of 26

Conditional Probability
Bayes’ Theorem

Definition and Properties
Independence
General Definition

Properties — analogous to Axioms of probability

1

0 ≤ Pr(A|B) ≤ 1.

2

Pr(S|B) = 1.

3

A1 ∩ A2 = ∅ ⇒ Pr(A1 ∪ A2 |B) = Pr(A1 |B) + Pr(A2 |B).

4

If A1 , A2 , . . . are all disjoint, then


Pr



Ai B
i=1

Prof. Jun Zhuang

Pr(Ai |B).

=
i=1

Fall 2014 EAS 305 Lecture Notes

Page 6 of 26

Conditional Probability
Bayes’ Theorem

Definition and Properties
Independence
General Definition

Independence — Any unrelated events are independent

Definition: A and B are independent iff
Pr(A ∩ B) = Pr(A)Pr(B).
A: It rains on Mars tomorrow.
B: Coin lands on H.
Example: If Pr(rains on Mars) = 0.2 and Pr(H) = 0.5, then
Pr(rains and H) = 0.1.
Note: If Pr(A) = 0, then A is indep of any other event.

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 7 of 26

Conditional Probability
Bayes’ Theorem

Definition and Properties
Independence
General Definition

Remark: Events don’t have to be physically unrelated to be indep. Example: Die. A = {2, 4, 6}, B = {1, 2, 3, 4}, A ∩ B = {2, 4}, so Pr(A) = 1/2, Pr(B) = 2/3, Pr(A ∩ B) = 1/3.
Pr(A)Pr(B) = 1/3 = Pr(A ∩ B) ⇒ A, B indep.

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 8 of 26

Conditional Probability
Bayes’ Theorem

Definition and Properties
Independence
General Definition

More natural interpretation of independence

Theorem
Suppose Pr(B) > 0. Then A and B are indep iff Pr(A|B) = Pr(A). Proof: A, B indep ⇔ Pr(A ∩ B) = Pr(A)Pr(B) ⇔
Pr(A ∩ B)/Pr(B) = Pr(A).
Remark: So if A and B are indep, the prob of A doesn’t depend on whether or not B occurs.

Prof. Jun Zhuang

Fall 2014 EAS 305 Lecture Notes

Page 9 of 26

Conditional Probability
Bayes’ Theorem

Definition and Properties
Independence
General Definition

Theorem
¯
A, B indep ⇒ A, B indep.
¯
Proof: Pr(A) = Pr(A ∩ B) + Pr(A ∩ B), so that
¯...
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