AP® Statistics · Unit 2 · Topic 2.7
The General Addition Rule
To find the chance that event A or event B happens, add their individual probabilities then subtract the chance both happen, so you don't count that overlap twice.
Picture two overlapping circles in a Venn diagram, one for A and one for B. When you add P(A) and P(B), you sweep through the overlapping middle region twice: once as part of A, once as part of B. That middle region is P(A and B), the outcomes where both events happen at the same time. To fix the double count, you subtract it back out once. That's the whole idea behind P(A or B) = P(A) + P(B) - P(A and B).
This rule works for any two events, whether or not they can happen together. If A and B can never happen at the same time, meaning they are mutually exclusive, then P(A and B) = 0 and the rule quietly simplifies to the simpler addition rule you probably saw first, P(A or B) = P(A) + P(B). So that earlier rule is really just a special case of this one.
This connects directly to the two-way tables and Venn diagrams used earlier in the unit, since P(A and B) is exactly the count or proportion sitting in the shared cell or overlapping region. It also sets up conditional probability and independence, where you'll use P(A and B) again but build it from P(A) times P(B given A) instead of reading it straight off a table.
Key terms
- General addition rule
- The formula P(A or B) = P(A) + P(B) - P(A and B), valid for any two events.
- Mutually exclusive
- Describes two events that cannot both happen at once, so P(A and B) = 0.
- Overlap
- The outcomes shared by both events, equal to P(A and B), which gets counted twice by simple addition.
- Venn diagram
- A picture of two overlapping circles used to visualize shared and separate outcomes between events.
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End of topic.