What Is Probability of A or B?
The probability of A or B is the chance that A happens, B happens, or both happen. It is also called the union of two events and written P(A ∪ B).
The addition rule gives it as P(A) + P(B) − P(A and B). Subtracting the overlap stops the shared outcomes from being counted twice. For mutually exclusive events the overlap is zero, so you simply add.
Enter P(A) and P(B), and say how the events relate. The calculator returns the union, the intersection and the remaining regions.
How to Use the Probability of A or B Calculator
- Enter P(A) and P(B).
- Tell the calculator how the events relate.
- Read the union, the overlap and the other regions.
Probability of A or B Formula
Addition rule
Independent events
Neither
Probability of A or B Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| P(A) | Probability of event A | |
| P(B) | Probability of event B | |
| P(A and B) | Probability that both happen | 0 if mutually exclusive, P(A)P(B) if independent |
| P(A or B) | Probability that at least one happens | P(A) + P(B) − P(A and B) |
Probability of A or B Worked Example
With the example values loaded in the calculator above, the steps are:
- Use the addition rule P(A or B) = P(A) + P(B) − P(A and B).
- P(A and B) = 0.12 (independent: P(A and B) = P(A) × P(B)).
- P(A or B) = 0.4 + 0.3 − 0.12 = 0.58.
- P(neither) = 1 − 0.58 = 0.42.
Result: P(A or B) = 0.58; P(A and B) = 0.12.
Worked example: P(A) = 0.4, P(B) = 0.3, independent
The overlap is 0.4 × 0.3 = 0.12. So P(A or B) = 0.4 + 0.3 − 0.12 = 0.58.
The chance that neither happens is 1 − 0.58 = 0.42, which also equals 0.6 × 0.7.
Mutually exclusive events
If A and B cannot happen together, the overlap is 0 and P(A or B) is just P(A) + P(B). Rolling a 2 or a 5 on one die gives 1/6 + 1/6 = 1/3.
Where Probability of A or B Is Used
Probability homework
Apply the addition rule without mixing up the overlap.
Risk of either event
Find the chance that at least one of two risks occurs.
Checking a Venn diagram
Confirm every region adds to the right total.
Common Probability of A or B Mistakes to Avoid
- Mutually exclusive and independent are different ideas.
- You can type probabilities as decimals, percentages or fractions.
Probability of A or B Terms and Definitions
- Union
- The event that A, B or both occur, written A ∪ B.
- Mutually exclusive
- Events that cannot happen at the same time.
Probability of A or B FAQs
How do you find the probability of A or B?
Add P(A) and P(B) and subtract P(A and B).
What if A and B are mutually exclusive?
Then P(A and B) = 0 and you simply add the two probabilities.
What if A and B are independent?
Use P(A and B) = P(A) × P(B) in the addition rule.
Are mutually exclusive events independent?
No. If one happens, the other cannot, so knowing one changes the chance of the other.
What does or mean in probability?
It is inclusive: A or B includes the case where both occur.
How do I find the chance of neither?
Subtract P(A or B) from 1.