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==Proof == | ==Proof == | ||
− | By Kolmogorov axioms, for pairwise disjoint events A<math>_1</math> and A<math>_2</math> in ''F'', <br/> | + | By Kolmogorov axioms, for pairwise disjoint events A<math>_1</math> and A<math>_2</math> in event space ''F'', <br/> |
<center><math>P(A_1\cup A_2) = P(A_1)+P(A_2)</math></center> | <center><math>P(A_1\cup A_2) = P(A_1)+P(A_2)</math></center> | ||
Revision as of 14:46, 15 October 2013
Theorem
Let A$ _1 $,A$ _2 $,...,A$ _n $ be n pairwise disjoint events in event space. Then
Proof
By Kolmogorov axioms, for pairwise disjoint events A$ _1 $ and A$ _2 $ in event space F,
Induction Hypothesis: Let A$ _1 $,A$ _2 $,...,A$ _n $ be pairwise disjoint events in F, i.e.
Then,
Now, I want to show that for pairwise disjoint events A$ _i $ i = 1,2,...,n+1
Let A$ _{n+1} $ ∈ F such that A$ _{n+1} $ is pairwise disjoint from events A$ _i $, i = 1,2,...,n.
Let
Note that B ∩ A$ _{n+1} $ = ø because if x ∈ A$ _{n+1} $, then x ∉ A$ _i $ ∀i = 1,2,...,n ⇒ x ∉ B. On the other hand if x ∈ B, then x ∈ A$ _i $ for exactly on i since A$ _i $'s are pairwise disjoint. A$ _{n+1} $ and A$ _i $ are also disjoint so x ∉ A$ _{n+1} $
We mentioned earlier that the probability of the union of two disjoint events is the sum of the probabilities of those events. Hence,
Thus by induction, we have that for a finite collection of pairwise disjoint events A$ _1 $ through A$ _n $ in F,
$ \blacksquare $