Independent Events Probability Calculator (AND / OR)
Combine two independent event probabilities: the chance both happen (AND), either happens (OR), and neither happens.
How it works
P(A and B) = P(A) × P(B) · P(A or B) = P(A) + P(B) − P(A and B) · P(neither) = (1 − P(A)) × (1 − P(B))
When two events are independent — one does not affect the other — the chance both happen is the product of their probabilities. The chance either happens uses the addition rule and subtracts the overlap so it is not counted twice. These only hold under independence; if the events influence each other you need conditional probabilities instead.
Worked example
If A has a 50% chance and B a 40% chance, both happen at 0.5 × 0.4 = 20%. Either happens at 0.5 + 0.4 − 0.2 = 70%, and neither happens at 0.5 × 0.6 = 30% — which correctly complements the 70%.
Frequently asked questions
What does independent mean here?
Two events are independent when the outcome of one does not change the probability of the other, like two separate coin flips. The multiplication rule for AND only applies in that case.
Why subtract P(A and B) in the OR formula?
Adding P(A) and P(B) counts the overlap — where both occur — twice. Subtracting P(A and B) once removes the double count, giving the true chance of at least one.
How do I get the chance of exactly one event?
Take P(A or B) and subtract P(A and B). That leaves the cases where one happens but not both — here 70% − 20% = 50%.
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This calculator is for educational purposes. Double-check important results before acting on them.