Definition D1
Two events are called disjoint when X ∩ Y = ∅. Events A₁, …, Aₙ are called mutually disjoint when every two distinct ones are disjoint.
Proof.
Two disjoint events do not occur simultaneously.convention
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Theorem T1
If A₁, …, Aₙ are mutually disjoint, then P(A₁ ∪ ⋯ ∪ Aₙ) = P(A₁) + ⋯ + P(Aₙ)
Proof.
(1)For n = 2 this is additivity itself.A2, D1
(2)Suppose it holds for n = k.assumption
(3)Let B = A₁ ∪ ⋯ ∪ A_k.convention
(4)B ∩ A_{k+1} = (A₁ ∩ A_{k+1}) ∪ ⋯ ∪ (A_k ∩ A_{k+1})(3), A4, A5
(5)Each term is empty, so B ∩ A_{k+1} = ∅, and B and A_{k+1} are disjoint.(4), assumption, D1
(6)P(B ∪ A_{k+1}) = P(B) + P(A_{k+1})(5), A2
(7)P(B) = P(A₁) + ⋯ + P(A_k)(2), (3)
(8)A₁ ∪ ⋯ ∪ A_{k+1} = B ∪ A_{k+1}(3), A5
(9)P(A₁ ∪ ⋯ ∪ A_{k+1}) = P(A₁) + ⋯ + P(A_{k+1})(6), (7), (8), E
(10)By induction, the statement holds for all n ≥ 2.(1), (9), A3
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Theorem T2
For two events not assumed disjoint, P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Proof.
(1)A ∪ B = (A∖B) ∪ (A∩B) ∪ (B∖A), and the three events are mutually disjoint.A6, D1
(2)P(A ∪ B) = P(A∖B) + P(A∩B) + P(B∖A)(1), T1
(3)A = (A∖B) ∪ (A∩B), and the two events are disjoint.A7, D1
(4)P(A) = P(A∖B) + P(A∩B)(3), A2
(5)B = (A∩B) ∪ (B∖A), and the two events are disjoint.A8, D1
(6)P(B) = P(A∩B) + P(B∖A)(5), A2
(7)P(A) + P(B) = P(A∖B) + P(A∩B) + P(B∖A) + P(A∩B)(4), (6), E
(8)P(A) + P(B) = P(A ∪ B) + P(A∩B)(2), (7), A9
(9)P(A ∪ B) = P(A) + P(B) − P(A∩B)(8), A10
(10)In P(A) + P(B), the term P(A∩B) appears twice. The subtracted term is not a correction but that duplication itself.(7), (8)
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Corollary C1
P(A ∪ B) ≤ P(A) + P(B), with equality if and only if P(A ∩ B) = 0.
Proof.
(1)P(A ∩ B) ≥ 0A1
(2)P(A ∪ B) = P(A) + P(B) − P(A∩B)T2
(3)P(A ∪ B) ≤ P(A) + P(B)(1), (2), A11
(4)By (2), the difference between the two values is P(A∩B), so equality holds only when P(A∩B) = 0.(2), (3)
(5)If the events are disjoint, then A ∩ B = ∅, and T2 reduces to additivity.(2), D1, A2
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