How to use this calculator
Use a binomial model when you repeat a trial a fixed number of times, each trial has two outcomes, and the success probability stays the same. The exact mode gives one success count. The cumulative modes add the probabilities of all counts in the selected range.
The method
Enter the values in the labeled fields, check the units or selected mode, and choose Calculate. The result includes the applicable working and a breakdown or visual when useful. Change an input and calculate again to update the result.
Number of trials (n): 10 Success probability (p): 0.5 Probability to calculate: Exactly k successes Successes (k): 5
Binomial probability: 24.609375%
P(X = 5)
Scope and limitations
The model assumes independent trials and a fixed success probability. It supports 0-2,000 trials and integer success counts. Extremely small tail probabilities can underflow in floating-point arithmetic.
Common questions
Does "at least" include k?
Yes. At least k means k or more, so the probability for exactly k is included.
What should I enter for probability?
Enter a number from 0 to 1. For a 20% success chance, enter 0.2, not 20.
References & further reading
External references provide background, not an endorsement of this implementation. See our methodology for numerical conventions and limitations.