Case 1
- P(A) — probability of event A
- 0.5
- P(B) — probability of event B
- 0.4
- P(A∩B) — probability of both
- 0.2
0.7000
Open with these valuesProbability Union Calculator
0.7000
Result: 0.7000Adding two probabilities counts the overlap twice, so the overlap has to come off once. A 50 % event and a 40 % event that coincide 20 % of the time give 70 %, not 90 %. Enter 0 for the overlap when the two events cannot both happen.
Worked examples
Case 2
- P(A) — probability of event A
- 0.3
- P(B) — probability of event B
- 0.5
- P(A∩B) — probability of both
- 0
0.8000
Open with these valuesCase 3
- P(A) — probability of event A
- 0.5
- P(B) — probability of event B
- 0.5
- P(A∩B) — probability of both
- 0.25
0.7500
Open with these valuesHow it's calculated
P(A∪B) = P(A) + P(B) − P(A∩B)
- StepEnter the probability of event A as a decimal between 0 and 1.
- StepEnter the probability of event B the same way.
- StepEnter how likely both happen together, or 0 if they cannot.
- ResultRead the chance that at least one of the two happens.
What this number means
The overlap is counted twice, so it comes off once
Outcomes in which both events happen belong to A and to B alike. Adding P(A) and P(B) counts that shared region twice, and subtracting P(A∩B) removes the double count exactly once.
Only mutually exclusive events take a 0
The subtraction falls away only when the two cannot both happen, leaving P(A∪B) = P(A) + P(B). For 0.3 and 0.5 that is 0.8000.
Independent is not the same as exclusive
Two independent halves still overlap: P(A∩B) = 0.25, and their union is 0.7500, not 1. Entering 0 for that overlap would claim the two can never coincide.
Contradictory inputs are pulled back into range
The overlap cannot exceed the smaller of P(A) and P(B), and the union cannot exceed 1. Numbers that break those rules describe events that cannot exist, and the nearest valid probability is shown instead.
Commonly misread
P(A) = 0.5 and P(B) = 0.4 make the chance of A or B 0.9.
That holds only if the two never coincide. With an overlap of 0.2 the union is 0.5 + 0.4 − 0.2 = 0.7.
The overlap only has to be subtracted for dependent events.
It has to come off whenever both events can happen together. Two independent halves overlap 0.25, and their union is 0.7500 rather than 1.
A union above 1 just means the events are very likely.
No probability exceeds 1, so those three numbers describe events that cannot exist. P(A) = 0.7 with P(B) = 0.6 needs an overlap of at least 0.3, which lands exactly on 1.
Reference table
| P(A), P(B), P(A∩B) | Case | P(A∪B) |
|---|---|---|
| 0.5, 0.4, 0.2 | overlapping events | 0.7000 |
| 0.3, 0.5, 0 | mutually exclusive | 0.8000 |
| 0.25, 0.15, 0.05 | small overlap | 0.3500 |
| 0.5, 0.5, 0.25 | two independent halves | 0.7500 |
| 0.7, 0.6, 0.3 | heavy overlap, certain union | 1.0000 |
Questions
How do I calculate the probability of a union?
Use the addition rule P(A∪B) = P(A) + P(B) − P(A∩B): add the two probabilities and subtract the chance that both happen. With P(A) = 0.5, P(B) = 0.4 and P(A∩B) = 0.2 the union is 0.7.
Why do you subtract the overlap?
Because the outcomes in which both events happen belong to A and to B alike. Adding P(A) and P(B) counts that shared region twice, and subtracting P(A∩B) removes the double count exactly once.
What if the two events are mutually exclusive?
Then they cannot both occur, their intersection is 0, and the rule simplifies to P(A∪B) = P(A) + P(B). Rolling a 1 or a 2 on one die is 1/6 + 1/6 = 1/3.
What does P(A∪B) mean?
It is the probability of the union of A and B: the chance that A happens, that B happens, or that both do. In plain language it is P(A or B), with an inclusive or that allows both at once.
What if my three numbers contradict each other?
The overlap cannot exceed the smaller of P(A) and P(B), and the union cannot exceed 1. Inputs that break those rules describe events that cannot exist, and the calculator falls back to the nearest valid probability rather than showing one above 1.
Sources and last check
Information, not professional advice.