Just as the difference rule gives rise to a formula for the probability of the complement of an event, so the addition and inclusion/exclusion rules give rise to formulas for the probability of the union of mutually disjoint events and for a general union of (not necessarily mutually exclusive) events.
To prove that for mutually disjoint events ,
for mutually disjoint sets.
The objective is to prove that for mutually disjoint events .
Let be the number of elements in the sample space .
By addition rule, for mutually disjoint sets.
Divide by .
To prove that for any events .
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