Two independent events
With P(A) = 0.5 and P(B) = 0.4, independent, P(A and B) = 0.20 and P(A or B) = 0.70.
Probability calculator
Result
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How it works
If A and B are independent (one happening tells you nothing about the other), P(A and B) is simply their product. If they are not independent, that intersection has to be measured or given directly, since knowing P(A) and P(B) alone is not enough to determine how much they overlap — enter it yourself and every other value is derived from that instead.
The normal-distribution mode answers a different kind of probability question: given a bell-curve distribution with a known mean and spread, what fraction of outcomes fall below (or above) a specific value.
Formulas used
Union
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Independent intersection
P(A ∩ B) = P(A) × P(B), only if independent
Worked examples
A probability under the normal distribution
With a mean of 0 and a standard deviation of 1, P(X < 1) = 84.13%.
Assumptions and limits
- All probabilities must be between 0 and 1.
- The normal-distribution mode assumes the underlying data genuinely follows a normal (bell-curve) distribution.
Frequently asked questions
What does it mean for two events to be independent?
Knowing whether one occurred gives you no information about whether the other did — flipping two separate coins is independent, but drawing two cards from a deck without replacement is not, since the first draw changes what is left.
What is the difference between union and intersection?
The intersection, P(A and B), is the chance both happen. The union, P(A or B), is the chance at least one happens — it is not simply P(A) + P(B), since that would double-count the outcomes where both occur.
What does the symmetric difference mean?
P(A or B, but not both) — exactly one of the two events happens, not neither and not both. It equals the union minus the intersection.
Why 84.13% for P(X < 1) with a standard normal distribution?
That is the classic "68-95-99.7 rule" in disguise: about 68% of a normal distribution falls within 1 standard deviation of the mean, split evenly, so roughly 34% lies between the mean and 1 standard deviation above it — adding the 50% below the mean gives 84%.
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