Addition rule is one of 3 rules formulas in the probability section of this library, and it is used at high school · ap level.
Why addition rule works
Adding the two probabilities counts every outcome in the overlap twice, once inside each event. Subtracting the overlap a single time brings the double-counted region back down to being counted once, which is exactly what the union should contain.
What each symbol means
$A\cup B$ means A or B; $A\cap B$ means both.
Addition rule: when it holds
Subtract the overlap once because it was counted twice.
When it stops applying
With three or more events this two-term version is not enough. An outcome lying in all three gets added three times and then removed three times, so it vanishes from the total and must be restored by one further addition at the end.
Addition rule: a worked example
If $.4,.5,$ and overlap $.2$, then union probability is $.7$.
The mistake to avoid
What people do: Adding the two probabilities without subtracting anything when the events can happen together.
Why it goes wrong: For drawing a heart or a face card, adding gives 13 out of 52 plus 12 out of 52, which is 25 out of 52. The three cards that are both get counted twice, so the true answer is 22 out of 52, or 11 out of 26.
Do this instead: Ask whether a single outcome could satisfy both descriptions. If it can, find how likely that overlap is and subtract it once.
Addition rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $A\cup B$ means A or B; $A\cap B$ means both.
- Check the conditions before substituting. Subtract the overlap once because it was counted twice.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Rules
- Level
- High school · AP
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where addition rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about addition rule
When can I simply add the two probabilities?
When the events are mutually exclusive, meaning no outcome satisfies both. The overlap is then zero, the subtraction changes nothing, and the rule collapses to plain addition.
What can I do if the overlap is not given?
You can still bound the answer. The union is at most the sum of the two probabilities and at least the larger of them, which is often enough to answer a yes-or-no question.
How does the rule extend to three events?
Start with the three individual probabilities, take away each overlap of two, then put the region common to all three back on. The signs keep alternating as further events join in.
Does the word or here include the possibility of both?
Yes. In probability, or is always inclusive, so A or B means at least one of them happens. Everyday speech often means one or the other but not both, which is a different event.