How to use the axioms of probability to derive the following result: For any events; A, B, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).?
Any help explaining this would be greatly appreciated. Thanks
You may assume the following: For any events A, B, P(A) = P(A ∩ B) + P(A ∩ B).
- DixonLv 71 month ago
The only axiom that is relevant is;
If A and B are mutually exclusive outcomes,
P(A ∪ B ) = P(A) + P(B)
If they are not mutually exclusive and we still want P(A ∪ B ) then by definition or inspection, if we start with P(A) then the additional area for the union is P(B) minus the overlap (because we already accounted for all A's).
Given the overlap is defined as A ∩ B then
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
This clearly works for mutually exclusive events too since P(A ∩ B) becomes zero.
The well know tips are
i∩tersection (overlap, both true)
U nion ( "shadow" area of the parts, logical OR)
I would add, always sketch the question if possible because the diagrams are nearly always easier to interpret.