Editing Probability axioms (section)

== Further consequences ==
Another important property is:

: <math>P(A \cup B) = P(A) + P(B) - P(A \cap B).</math>

This is called the addition law of probability, or the sum rule.
That is, the probability that an event in ''A'' ''or'' ''B'' will happen is the sum of the probability of an event in ''A'' and the probability of an event in ''B'', minus the probability of an event that is in both ''A'' ''and'' ''B''. The proof of this is as follows:

Firstly,

:<math>P(A\cup B) = P(A) + P(B\setminus A)</math>. ''(by Axiom 3)''

So,

:<math>P(A \cup B) = P(A) + P(B\setminus  (A \cap B))</math> (by <math>B \setminus A = B\setminus  (A \cap B)</math>).

Also,

:<math>P(B) = P(B\setminus (A \cap B)) + P(A \cap B)</math>

and eliminating <math>P(B\setminus (A \cap B))</math> from both equations gives us the desired result.

An extension of the addition law to any number of sets is the [[inclusion–exclusion principle]].

Setting ''B'' to the complement ''A<sup>c</sup>'' of ''A'' in the addition law gives

: <math>P\left(A^{c}\right) = P(\Omega\setminus A) = 1 - P(A)</math>

That is, the probability that any event will ''not'' happen (or the event's [[Complement (set theory)|complement]]) is 1 minus the probability that it will.