Quick start
- Enter P(A) and P(B) as percentages.
- Leave P(A and B) blank only when the events are independent.
- Enter P(A and B) when you already know the overlap.
- Press Calculate probability.
- Review P(A or B), P(A and B), not A, not B, and the formula steps.
- Fix the inputs if the calculator says the overlap or union is impossible.
Best uses
The Probability Calculator is most useful when you already know the event chances and need the overlap, union, complements, or a quick consistency check.
- Find the chance that A or B happens when you know both event probabilities.
- Enter a known overlap when two events can happen together.
- Leave the overlap blank when the events are independent.
- Calculate complements such as not A and not B.
Quick answer
If you enter P(A) = 40% and P(B) = 25% and leave P(A and B) blank, the calculator treats the events as independent. It multiplies 40% by 25% to get a 10% overlap, then uses that overlap in the union formula.
The result is P(A or B) = 55%. That means the chance of A, B, or both happening is 55% under the numbers you entered.
What each input means
P(A) is the chance that event A happens. P(B) is the chance that event B happens. The optional P(A and B) field is the chance that both happen together.
All three fields use percentages. Enter 12.5 for 12.5%, not 0.125. Each value must be between 0% and 100%.
- Use P(A) for the first event, such as rain tomorrow.
- Use P(B) for the second event, such as a delivery arriving late.
- Use P(A and B) only when you know the overlap or joint chance.
Independent events versus known overlap
Independent events do not change each other. If A happening does not affect B, leaving the overlap blank is reasonable because the calculator can estimate P(A and B) with multiplication.
If the events are connected, do not leave the overlap blank unless the problem says they are independent. Type the known overlap into P(A and B) instead.
- Independent example: P(A)=40% and P(B)=25% gives P(A and B)=10%.
- Known-overlap example: P(A)=60%, P(B)=30%, and P(A and B)=15% gives P(A or B)=75%.
- Mutually exclusive example: if A and B cannot happen together, enter P(A and B)=0%.
How the union formula works
P(A or B) means A happens, B happens, or both happen. The calculator uses P(A or B) = P(A) + P(B) - P(A and B).
The subtraction matters because the overlap is included once in P(A) and once in P(B). Subtracting P(A and B) keeps the same outcomes from being counted twice.
- For 60%, 30%, and 15%, the union is 60% + 30% - 15% = 75%.
- For 20%, 30%, and 0%, the union is 20% + 30% - 0% = 50%.
- For 80%, 70%, and 60%, the union is 80% + 70% - 60% = 90%.
Complements and what they tell you
The complement is the chance that an event does not happen. If P(A) is 40%, not A is 60%. If P(B) is 25%, not B is 75%.
Complements are useful when a problem is easier to think about from the other side. Sometimes "at least one" is easier to solve by finding "none" first, then subtracting from 100%.
Mistakes and limits
The calculator rejects impossible probability combinations. P(A and B) cannot be larger than P(A) or P(B), and P(A or B) cannot be more than 100%.
The tool does not count dice rolls, cards, or combinations for you. First count favorable and total outcomes, turn that into a percentage, and then enter the probability. If the counting step is the hard part, use the Permutation and Combination Calculator first.
- Do not leave the overlap blank just because you do not know it.
- Do not mix percentages and decimals in the input fields.
- Do not treat the result as a promise about what will happen.
Worked examples for Probability Calculator
P(A and B)=10%; P(A or B)=55%
P(A or B)=75%
P(not A)=60%
P(A or B)=50%
P(A or B)=90%
P(A and B)=2.5%; P(A or B)=30%
FAQ in plain language
What does P(A or B) mean?
P(A or B) is the chance that A happens, B happens, or both happen.
What does P(A and B) mean?
P(A and B) is the overlap: the chance that both events happen together.
When should I leave P(A and B) blank?
Leave it blank only when the events are independent or your problem tells you to multiply P(A) by P(B).
What if I already know the overlap?
Enter the known P(A and B) value. That is more accurate than leaving the field blank and letting the calculator assume independence.
What if A and B are mutually exclusive?
Enter 0 for P(A and B). Then the calculator adds P(A) and P(B) because the events cannot happen together.
Can I enter decimal percentages?
Yes. Enter 12.5 for 12.5%. Do not enter 0.125 unless you mean 0.125%.
Why did the calculator reject my inputs?
The inputs may contradict each other. For example, the overlap cannot be larger than either event, and the union cannot be greater than 100%.
Sources
Use these if you want to compare the formula, inputs, or limits with a trusted outside explanation.
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