If a is a member of B, this is denoted a ∈ B
If every member of set A is also a member of set B, then A is said to
be a subset of B, written A ⊆ B (A is said to be contained in B).
Stat 330 (Spring 2015): slide set 2
A ∩ B = {ω | ω ∈ A and ω ∈ B}
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3. Intersection (∩): An intersection of events is an event consisting of the
common outcomes in these events.
A ∪ B = {ω | ω ∈ A or ω ∈ B}
2. Union (∪): A union of events is an event consisting of all the outcomes
in these events.
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1. Review symbols ∈, ∈,
/ ⊂, ⊆, ⊃, ⊇,.
For e.g.
Sets and Venn Diagrams
Last update: January 13, 2015
Stat 330 (Spring 2015)
Slide set 2
To be able to work with probabilities, in particular, to be able to compute
probabilities of events, a mathematical foundation is necessary.
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Ai ∩ Aj = ∅ for any i = j.
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8. Mutually exclusive sets: Sets A1, A2, . . . are mutually exclusive if any
two of these events are disjoint:
A∩B =∅
7. Disjoint sets: Sets A, B are disjoint if their intersection is empty:
∅⊆A
6. Empty Set ∅ is a set having no elements, usually denoted by {}
The empty set is a subset of every set:
(A ∪ B) = Ā ∩ B̄
5. Demorgan’s Law:
Ā = {ω | ω ∈
/ A}
4. Complement (Ā): A complement of an event A is an event that occurs
when event A does not happen.
Stat 330 (Spring 2015): slide set 2
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We write: P (C) = 0.9
From our experience with the network provider, we can decide that the
chance that the next message gets through is 90 %.
Consider the Event C (a successful transmission) defined earlier.
Stat 330 (Spring 2015): slide set 2
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Example:
Probability
Stat 330 (Spring 2015): slide set 2
Kolmogorov’s Axioms (continued)
Example:
Draw a single card from a standard deck of playing cards: Ω = {red, black}
Model 2
P (Ω) = 1
P (red) = 0.3
P (black) = 0.7
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Mathematically, both schemes are equally valid. But, of course, our real
world experience would favor to pick model 1 over model 2 as the ‘correct’
model.
Model 1
P (Ω) = 1
P (red) = 0.5
P (black) = 0.5
Corollary: For any A, P (A) ≤ 1.
3. If A ⊂ B, then P (A) ≤ P (B).
P (A∪B) = P (A)+P (B)−P (A∩B)
2. Addition Rule of Probability
Corollary: P (∅) = 0
1. Probability of the Complementary Event: P (A) = 1 − P (A)
Let A, B ⊂ Ω.
Useful Consequences of Kolmogorov’s Axioms:
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Stat 330 (Spring 2015): slide set 2
Stat 330 (Spring 2015): slide set 2
i
(iii) if A1, A2, . . . are (possibly, infinite many) mutually exclusive events (i.e.
Ai ∩ Aj = ∅ for all i = j) then
P (Ai).
P (A1 ∪ A2 ∪ . . .) = P (A1) + P (A2) + . . . =
(ii) P (Ω) = 1.
(i) 0 ≤ P (A) ≤ 1 for all A
A system of probabilities (a probability model) is an assignment of numbers
P (A) to events A ⊂ Ω (requires further mathematical complications for the
events that we can define probability on) in such a manner that
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These are the basic rules of operation of a probability model
• every valid model must obey these,
• any system that does, is a valid model
Whether or not a particular model is realistic or appropriate for a specific
application is another question.
Two different, equally valid probability models are:
Stat 330 (Spring 2015): slide set 2
To be able to work with probabilities properly - to compute with them - one
must lay down a set of postulates:
Kolmogorov’s Axioms
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Sets and Venn Diagrams (contd.)
Stat 330 (Spring 2015): slide set 2
4. What is the probability that only the phone is up?
3. What is the probability that we fail to connect?
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2. What is the probability that both the phone and the network are down?
1. What is the probability that the phone is up or the network is up?
P ( phone up and network up ) = .55.
P ( network up ) = .6, and
P ( phone up ) = .9
We attempt to sign on to AOL using dial-up. We connect successfully if
and only if the phone number works and the AOL network works. Assume
Example: Using Kolmogorov’s Axioms
Stat 330 (Spring 2015): slide set 2
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P (Ā ∪ B̄) = (1 − .9) + (1 − .6) − .05 = .1 + .4 − .05 = .45
P ( phone down or network down) = P (Ā∪ B̄) = P (Ā)+P (B̄)−P (Ā∩ B̄)
3) What is the probability that we fail to connect?
= 1 − .95 = .05
P ( phone down and network down) = P (Ā ∩ B̄) = P (A ∪ B)
2) What is the probability that both the phone and the network are down?
P ( phone up or network up) = P (A ∪ B) = 0.9 + 0.6 − 0.55 = 0.95
1) What is the probability that the phone is up or the network is up?
Let A ≡ phone up; B ≡ network up
Solution