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Probability
01. Sample Space
郭俊利
2009/02/27
1
Outline
Probability
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Sample space
Probability axioms
Conditional probability
Independence
1.1 ~ 1.5
2
Introduction
Probability

What is probability?
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Time
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Space
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Frequency
Area
Examples
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Weather forecast ^^
Cancer prediction ^^
Lottery
>”<
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Sets
Probability
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Sample space
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List of all possible outcomes
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S1 = {H, T} (H = head; T = tail)
S2 = { (H, H), (H, T), (T, H), (T, T) }
Event
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A subset of the sample space
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Basic Laws
Probability

Axioms:
1. 0 ≦ P(A) ≦ 1
2. P(S) = P(universe) = 1
3. If A∩B = Ø, then P(A∪B) = P(A) + P(B)
4. If A∩B ≠ Ø, …
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Real Laws
Probability
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P(A∪B) = P(A) + P(B) – P(A∩B)
= P(A) + P(AC∩B)
P(A∪B∪C) = P(A) + P(B) + P(C) – P(A∩B) –
P(B∩C) – P(C∩A) + P(A∩B ∩C)
P(A∪B∪C) = P(A) + P(AC∩B) + P(AC∩BC∩C)
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Example 1
Probability

Bonferroni’s inequality
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P(A∩B) ≧ P(A) + P(B) – 1
P(A1∩A2∩…∩An) ≧ P(A1) + P(A2) + … +
P(An) – (n – 1)
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Example 2
Probability
P(A|B) =

P(A∩B)
P(B)
Given that the two dice land on different
numbers, find the conditional probability that
at least one die roll is a 6.
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Multiplication Rule
Probability
P(A|B) =


P(A∩B)
P(B)
P(A∩B) = P(B) P(A|B)
P(A∩B∩C) = P(A) P(B|A) P(C|A∩B)
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Example 3
Probability
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Independence
Probability


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P(A∩B) = P(B) P(A|B)
If A is independent of B, P(A) = P(A|B)
P(A∩B) = P(B) P(A)
If A is disjoint of B, then A is
independent of B?
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Example 4
Probability
36
2
3
3
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Example 5
Probability

You enter a chess tournament where your
probability of winning a game is 0.3 against
half the players, 0.4 against a quarter of the
players, and 0.5 against the remaining
quarter of the players. You play a game
against a randomly chosen opponent.


What is the probability of winning?
Suppose that you win. What is the probability that
you had an opponent of 3rd type?
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Conditional Independence
Probability
P(A∩B | C) =
P(A∩B∩C)
P(C)
=
P(C) P(B|C) P(A|B∩C)
P(C)
= P(B|C) P(A|B∩C)

P(A∩B | C) = P(A|C) P(B|C)
P(A|B∩C) = P(A|C)
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Example 6
Probability
H1 = {1st toss is a head}
H2 = {2nd toss is a head}
D = {the two tosses have different results}
P(H1|D) = ½; P(H2|D) = ½; P(H1∩H2 | D) = 0

H1 and H2 are independent, but not
conditionally Independent
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Example 7
Probability
H1 = {1st toss is a head}
H2 = {2nd toss is a head}
D = {the two tosses have different results}

H1
H2
Are H1, H2 and D
independent?
D
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Example 8
Probability
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Let A and B be independent.
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Are A and BC independent?
Are AC and BC independent?
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Example 9
Probability
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Let A, B , C be independent.
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Prove that A and B are conditionally
Independent given C.
P(A∩B | C) = P(A|C) P(B|C)
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Example 10
(1/2)
Probability
0.8
0.9
0.9
C
0.95
A
E
F
0.85
B
0.95
0.75
D
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Example 10
(2/2)
Probability
0.8
0.9
0.9
C
0.95
A
E
F
0.85
B
0.95
0.75
D
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