Download Basic properties of probability

Basic properties of probability
Math 308
Definition: Let S be a sample space. A probability on S is a real valued function P ,
P : {Events} → R,
satisfying:
1. P (A) ≥ 0 for any event A.
2. P (S) = 1.
3. If A1 , A2 , . . . are mutually exclusive events (m.e.e) then
P(
∞
[
Ai ) =
i=1
∞
X
P (Ai ),
i=1
where by m.e.e. we mean Ai ∩ Aj = ∅ when i 6= j.
Basic properties of probability:
1. P (∅) = 0.
2. Let A1 , A2 , . . . An be m.e.e., then
P(
n
[
Ai ) =
i=1
n
X
P (Ai ).
i=1
3. P (A0 ) = 1 − P (A).
4. If A ⊂ B then P (A) ≤ P (B).
In particular, if B is S, we get 0 ≤ P (A) ≤ 1 for any event A.
5. Let B1 , B2 , . . . be m.e.e. such that S = ∪∞
i=1 Bi , that is, the Bi ’s form a partition of S.
Then for any event A,
∞
X
P (A) =
P (A ∩ Bi ).
i=1
6. It follows from (5) that for any event B, since S = B ∪ B 0 is a partition of S, then
P (A) = P (A ∩ B) + P (A ∩ B 0 ).
In particular, since A ∩ B 0 = A \ B (draw the Venn diagram), we have
P (A \ B) = P (A) − P (A ∩ B).
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7. For any events A and B,
P (A ∪ B) = P (A) + P (B) − P (A ∩ B).
8. (probability of a finite union)
P(
n
[
i=1
Ai ) =
n
X
i=1
P (Ai )−
X
i1 <i2
X
P (Ai1 ∩Ai2 )+
P (Ai1 ∩Ai2 ∩Ai3 )−· · ·+(−1)n+1 P (∩ni=1 Ai ).
i1 <i2 <i3
In particular,
P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (A ∩ C) − P (B ∩ C) + P (A ∩ B ∩ C).
9. Using the basic properties (and Venn diagrams) you can find formulas for probabilities of
other operations on sets. For example, if A and B are events, then the probability that event
A occur or B occur, but not both is
P ((A ∪ B) \ (A ∩ B)) = P ((A \ B) ∪ (B \ A)) = P (A) + P (B) − 2P (A ∩ B).
Note that the last equality follows from property (2) since (A \ B) and (B \ A) are m.e.e.
(we also used property (6)).
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