How this calculator works
The multiplication rule uses a conditional probability in general. Independence is the special case in which P(B | A) equals P(B). The marginal probabilities also constrain the overlap: it cannot exceed either event and cannot be so small that their union exceeds one. A conditional value supplied for an impossible A is undefined and rejected. Independence must come from the experiment or model; knowing only two marginal probabilities does not establish it.
Formula
P(A and B) = P(A)P(B | A). For independent events: P(A and B) = P(A)P(B). P(A or B) = P(A) + P(B) - P(A and B).
Worked example
If independent events have probabilities 40% and 50%, both occur with probability 20%. If P(A) = 20%, P(B) = 30% and P(B | A) = 50%, their joint probability is 10% and their union is 40%.
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Frequently asked questions
Can I always multiply P(A) by P(B)?
Only with independence. Dependence changes the relevant conditional probability.
What bounds must the overlap satisfy?
It lies between max(0, P(A) + P(B) - 1) and min(P(A), P(B)).
Does joint probability mean either event?
Joint probability here means both events. The union output gives the chance of at least one.