How this calculator works
Exact probability counts the trial sequences that contain the requested number of successes, then weights each sequence by its probability. Inclusive intervals sum from the lower count through the upper count. Counts above n have zero mass. Zero trials and success chances of zero or one are valid degenerate distributions.
Formula
P(X=k) = C(n,k)p^k(1-p)^(n-k); E[X] = np; Var[X] = np(1-p).
Worked example
For four fair trials, exactly two successes have probability 6/16 = 0.375. At most two successes have probability 11/16 = 0.6875.
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Frequently asked questions
Can the success chance change between trials?
That needs a different model. This binomial calculation assumes one constant chance.
How are interval endpoints treated?
Both counts are included. An interval from 2 to 4 means X = 2, 3 or 4.
What does the quantile return?
The smallest count whose cumulative probability reaches the requested level.