Jan. 14
1.2 Properties of Probability
Event: A subset of the sample space (technical restrictions
unimportant for now)
Often use early-alphabet capital letters (e.g., A, B, C) for events.
Difference between an outcome and an event?
After an experiment, event A has occurred if . . .
Jan. 14
1.2
Properties of Probability
Algebra of sets and Venn diagrams
Null event and full sample space: ∅ and S
Subset notation: A ⊂ B or, alternatively, B ⊃ A or A implies B
Union: A ∪ B
Intersection: A ∩ B
Complement: A0 or, alternatively, Ac
Special random variable: The indicator IA or I(A) or 1A
Jan. 14
1.2 Properties of Probability
Mutually exclusive events: Have empty pairwise intersection
Exhaustive events: Have union equal to S
Partition: A group of mutually exclusive and exhaustive events
Jan. 14
1.2
Properties of Probability
An example: Roll two dice, so S has 36 elements or outcomes.
Let A = {s ∈ S : sum of dice results in s is even}
B = {first die is even}
(using abbreviated notation)
C = {second die is ≥ 5}
D = {sum is prime}
What is A ∩ D?
(B ∪ C 0 ∪ A)0 ?
(A0 ∩ B) ∪ (A ∩ B)?
Jan. 14
Appendix A.1
Algebra of sets
Jan. 14
Commutative property: A ∪ B = B ∪ A
Appendix A.1
Algebra of sets
Commutative property: A ∩ B = B ∩ A
Associative property: (A ∪ B) ∪ C = A ∪ (B ∪ C)
Associative property: (A ∩ B) ∩ C = A ∩ (B ∩ C)
Distributive property: A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Distributive property: A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
DeMorgan’s Law: (A ∪ B)0 = A0 ∩ B 0 or, more generally,
DeMorgan’s Law: (A ∩ B)0 = A0 ∪ B 0 or, more generally,
n
[
i=1
Jan. 14
!0
Ai
=
n
\
i=1
1.2 Properties of Probability
Probability: Real-valued set function, P, satisfying:
P(A) ≥ 0
(Nonnegativity)
If A is the whole sample space S then P(A) = 1
Whenever A1 , A2 , . . . are mutually exclusive,
!
∞
∞
[
X
P
Ai =
P(Ai )
(Countable additivity)
i=1
i=1
n
\
A0i
!0
Ai
i=1
Jan. 14
1.2
=
n
[
A0i
i=1
Properties of Probability
Some theorems:
(1) For any A, P(A) = 1 − P(A0 ).
(2) P(∅) = 0.
(3) If A implies B (i.e., A ⊂ B), then P(A) ≤ P(B).
(4) For any A, P(A) ≤ 1.
(5) For any A and B,
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
(5∗ ) For any A and B and C,
P(A ∪ B ∪ C) = P(A) + P(B) + P(C)
−P(A ∩ B) − P(A ∩ C) − P(B ∩ C)
+P(A ∩ B ∩ C)
(5∗∗ )
For any A1 , . . . , An ,
!
n
n
XX
[
X
P
Ai
=
P(Ai ) −
P(Ai ∩ Aj ) + · · ·
i=1
i=1
i<j
+(−1)n+1 P(A1 ∩ · · · ∩ An )
Jan. 14
Desired Outcomes
Students will be able to:
apply set notation to simplify expressions involving events;
apply Venn diagrams to the same;
prove simple facts based on the three axioms of probability;