Sample space Ω
LECTURE 1
• Readings: Sections 1.1, 1.2
• “List” (set) of possible outcomes
• List must be:
– Mutually exclusive
Lecture outline
– Collectively exhaustive
• Probability as a mathematical framework for:
• Art: to be at the “right” granularity
– reasoning about uncertainty
– developing approaches to inference problems
• Probabilistic models
– sample space
– probability law
• Axioms of probability
• Simple examples
Sample space: Discrete example
Sample space: Continuous example
• Two rolls of a tetrahedral die
Ω = {(x, y) | 0 ≤ x, y ≤ 1}
– Sample space vs. sequential description
1
4
Y = Second
y
1,1
1,2
1,3
1,4
1
2
3
roll
3
2
1
1
1
2
3
X = First roll
4
4
4,4
1
x
Probability axioms
Probability law: Example with finite sample
space
• Event: a subset of the sample space
4
• Probability is assigned to events
Y = Second 3
roll
Axioms:
2
1. Nonnegativity: P(A) ≥ 0
1
1
2. Normalization: P(Ω) = 1
2
3
4
X = First roll
3. Additivity: If A ∩ B = Ø, then P(A ∪ B) = P(A) +
P(B)
• Let every possible outcome have probability 1/16
– P((X, Y ) is (1,1) or (1,2)) =
• P({s1, s2, . . . , sk }) = P({s1}) + · · · + P({sk })
– P({X = 1}) =
= P(s1) + · · · + P(sk )
– P(X + Y is odd) =
– P(min(X, Y ) = 2) =
• Axiom 3 needs strengthening
• Do weird sets have probabilities?
Discrete uniform law
Continuous uniform law
• Let all outcomes be equally likely
• Then,
P(A) =
• Two “random” numbers in [0, 1].
y
1
number of elements of A
total number of sample points
• Computing probabilities ≡ counting
1
• Defines fair coins, fair dice, well-shuffled card decks
x
• Uniform law: Probability = Area
– P(X + Y ≤ 1/2) = ?
– P( (X, Y ) = (0.5, 0.3) )
2
Probability law: Ex. w/countably infinite sample
space
• Sample space: {1, 2, . . .}
– We are given P(n) = 2−n, n = 1, 2, . . .
– Find P(outcome is even)
p
1/2
1/4
1/8
1
2
3
1/16
…..
4
P({2, 4, 6, . . .}) = P(2) + P(4) + · · · =
1
1
1
1
+ 4 + 6 + ··· =
22
3
2
2
• Countable additivity axiom (needed for this calcu­
lation):
If A1, A2, . . . are disjoint events, then:
P(A1 ∪ A2 ∪ · · · ) = P(A1) + P(A2) + · · ·
3
MIT OpenCourseWare
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6.041SC Probabilistic Systems Analysis and Applied Probability
Fall 2013
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