OR, exclusive events and Venn diagrams
The word or combines routes. First decide whether the routes can happen together; that tells you whether an overlap must be corrected.
Mutually exclusive means no shared outcome
- means A and B: the shared outcomes in both events.
- means A or B, including the possibility that both happen.
- Events A and B are mutually exclusive when both cannot happen in one trial.
- Their intersection is empty: . The symbol means the empty set: there is no shared outcome.
- Therefore and their probabilities can be added directly.
Worked example
No overlap on one die
Rolling 1 and rolling 6 are mutually exclusive results of one die roll.
OR adds, then removes double-counting
Adding and counts the overlap twice. Subtract it once. For mutually exclusive events, the overlap is zero and the rule reduces to simple addition.
Worked example
Two events with overlap
, and .
OR describes a union. If the routes overlap, subtract the shared part once.
Examiner note
Fill a Venn diagram from the middle out
Fill a Venn diagram from the middle
both first → only regions → neither last
The rectangle is the universal set: the whole group being considered. Values outside the circles are in neither event.
- Write the overlap first. A stated “both” value belongs in the intersection.
- Find each only-region. Subtract the overlap from each whole-circle total.
- Find neither last. Subtract the complete union from 1, or from the stated population total.
Worked example
Desktop, laptop or neither
50% own a desktop, 60% own a laptop and 15% own both. The only regions are 0.35 and 0.45, so the union is 0.95.
Common mistake
Key idea
In a group, 42% study French, 35% study Spanish and 18% study both. Find the probabilities of studying exactly one language and studying neither.
Show worked answer
French-only is 0.24 and Spanish-only is 0.17, so exactly one is 0.41. The union is .