Enter the probability of each. You get both happening, either happening, exactly one, neither, and P(A given B).
This calculator treats A and B as independent — one happening tells you nothing about the other. Coin flips and dice rolls qualify. A great many real situations do not.
Rain today and rain tomorrow are not independent. Two loans defaulting in the same recession are not independent. Underestimating exactly this correlation is a large part of what made the 2008 mortgage crisis worse than the models predicted: individually unlikely defaults were treated as unrelated when they shared a common cause.
If your events are dependent, P(A and B) is P(A) × P(B given A), and you need the conditional probability rather than the raw one.
With P(A) = 0.5 and P(B) = 0.4, adding gives 0.9. But that counts the 0.2 where both occur twice — once inside A, once inside B. Subtracting the overlap gives the correct 0.7.
The trap sharpens with larger probabilities: two events at 0.7 each would sum to 1.4, which is impossible. Any probability above 1 means the overlap was not removed.
That one event happening does not change the odds of the other. Coin flips are independent; weather on consecutive days is not. The formulas here assume independence, and give wrong answers if the events are actually linked.
Because outcomes where both happen sit inside A and inside B, so adding the two probabilities counts them twice. Subtracting the overlap once corrects it. Without the subtraction you can get a probability above 1, which is impossible.
Take 1 minus the probability of none. For two events that is 1 − (1−P(A))(1−P(B)). It is almost always easier than adding up every way at least one could happen, and the advantage grows with more events.
The probability of A given that B has already happened: P(A and B) ÷ P(B). For independent events it simply equals P(A), since B tells you nothing. When it differs from P(A), the events are dependent by definition.
For counting the arrangements behind a probability, use the Permutation and Combination Calculator. Descriptive statistics on a dataset are handled by the Statistics Calculator and Standard Deviation Calculator, and for sampling see the Sample Size Calculator.