Combining two probabilities is not always as simple as adding or multiplying — this works out both the “and” and “or” cases for two independent events, and shows why the “or” case needs a subtraction most people leave out.
How it works
Combining two independent events
P(A and B) = P(A) × P(B) · P(A or B) = P(A) + P(B) − P(A and B)
“Independent” means A happening does not change the chance of B happening — this calculator assumes that throughout.
Why “or” is a subtraction, not just an addition
If two events could both happen, adding their probabilities straight counts the case where both happen twice — once inside each individual probability. Subtracting P(A and B) removes that double-count exactly once. Two coin flips both landing heads, and the second flip landing heads on its own, overlap in the “both heads” outcome; counting it twice would overstate the combined probability of “either happening.”
Why this only works for independent events
P(A and B) = P(A) × P(B) is only true when A happening does not change B’s chance. Drawing two cards from a deck without replacement are not independent — the first draw changes what remains for the second — and multiplying their plain probabilities gives a wrong answer for that case. This calculator assumes independence throughout; a dependent-events question needs a different calculation.
How to use this calculator
- Enter the probability of event A.
- Enter the probability of event B.
- Read both the “and” and “or” combined probabilities.
Frequently asked questions
What if my two events are not independent?
This calculator will give a wrong answer for the “and” case specifically — dependent events need the actual conditional probability of B given A, not B’s probability alone, which this page does not ask for.
Why does P(A or B) equal 100% when P(A) is 100%?
Because if A is certain, “A or B” is certain too — A alone already guarantees the “or” condition is met, regardless of what B is.
Can P(A and B) ever be bigger than either P(A) or P(B) alone?
No — for independent events, requiring both to happen is always at least as restrictive as requiring just one, so P(A and B) can never exceed the smaller of P(A) and P(B).
How is this different from conditional probability?
Conditional probability asks “what is the chance of B, given that A already happened” — a different question that can have a different answer when events are not independent. This calculator only handles the independent case.
Why do dice and card problems trip people up with this formula?
Because it is easy to assume independence when it does not actually hold — drawing cards without replacement, or events tied to the same physical outcome, are common places where the independence assumption silently breaks and this formula stops applying.
