Independent Joint Probability Calculator
Multiply the probabilities of independent events for the joint P(A and B) = P(A) × P(B), and get P(A or B) and “neither” too.
How it works
P(A ∩ B) = P(A) · P(B) · P(A ∪ B) = P(A) + P(B) − P(A)·P(B)
Two events are independent when one happening does not change the probability of the other, and for independent events the joint probability is just the product. The union then uses the general addition rule, subtracting the product so the overlap is not double-counted. Independence is a strong assumption — it must be justified by the setup, not assumed just because two events feel unrelated.
Worked example
A fair coin lands heads (P = 50%) and a spinner lands red (P = 40%), independently. P(heads and red) = 0.50 × 0.40 = 0.20 (20%), and P(heads or red) = 0.50 + 0.40 − 0.20 = 0.70.
Frequently asked questions
When can I just multiply the probabilities?
Only when the events are independent — the outcome of one has no effect on the other, like separate coin flips or dice. For dependent events you must use conditional probabilities.
Why subtract the product for P(A or B)?
The outcomes where both A and B happen get counted in P(A) and again in P(B). Subtracting P(A and B) removes that double count so the union is correct.
How is this different from mutually exclusive events?
Exclusive events never occur together, so P(A and B) = 0. Independent events can occur together; their joint probability is the product of the two.
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This calculator is for educational purposes. Double-check important results before acting on them.