Kolmogorov’s Axioms
Defn. A probability measure (p.m.) is a function P () such that
(i) P (A) ≥ 0 for A ⊂ Ω
(ii) P (Ω) = 1
(iii) If A1, . . . , An are mutually disjoint (Ai ∩ Aj = ∅ for i 6= j), then
P (∪∞
i=1 Ai ) =
∞
X
P (Ai)
i=1
Comments:
• Every valid probability measure must obey the axioms.
• If a funciton obeys the axioms, then the function is a valid probability measure.
• Whether or not a particular model is realistic or appropriate in
a particular situation is another question entirely...
Example: Draw a card from a standard deck, and record the
color.
Ω = {red, black}
Two mathematically valid probability models are in the table below. (You can
check that both specifications satisfy the axioms.)
P (Ω)
P (red)
P (black)
Model 1
1
.5
.5
Model 2
1
.3
.7
Although both models are legitimate from a mathematical standpoint, only Model
1 is appropriate for this scenario.
Useful Consequences of Kolmogorov’s Axioms:
Let A, B ⊂ Ω.
1. P (A) = 1 − P (A).
Corollary. P (∅) = 0.
2. Addition Rule of Probability:
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
3. If A ⊂ B, then P (A) ≤ P (B).
Corollary. For any A, P (A) ≤ 1.
Two handy inequalities are...
1. Bonferroni’s Inequality: P (A ∩ B) ≥ 1 − P (A) − P (B)
P
2. Boole’s Inequality: P (∪ni=1 Ai ) ≤ ni=1 P (Ai )
Here is some space for you to verify the three main results above using the axioms...
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Example: Using Kolmogorov’s Axioms
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 is up. Assume
P ( phone up ) = .9
P ( network up ) = .6, and
P ( phone up and network up ) = .55.
(a) What is the probability that we fail to connect?
(b) What is the probability that the phone is up or the network is up?
(c) What is the probability that both the phone and the network are down?
(d) What is the probability that only the phone is up?
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