Binomial Probability Step-by-Step Calculator
Calculate binomial probability with step-by-step support for exactly, at most, at least, and between events. The page is designed for students who need to know which probabilities to add, not just the final decimal.
When a binomial probability calculator is the right tool
A binomial problem has a fixed number of trials, only two outcomes per trial, the same probability of success on each trial, and independent trials. OpenStax lists these conditions in its exact lesson on the binomial distribution. This calculator is built around those conditions because using the binomial formula on the wrong kind of problem gives a polished but wrong answer.
Students search for binomial probability when a problem says things like “exactly 4 successes,” “at most 3,” “at least 6,” or “between 2 and 5.” This calculator handles each of those forms and shows what is being added behind the scenes.
Binomial probability formula
The combination term C(n, x) counts how many ways the successes can appear among the trials. The probability part pˣ(1 − p)ⁿ⁻ˣ handles the chance of that particular success/failure pattern.
Exactly, at most, at least, and between
“Exactly” uses one binomial probability. “At most” adds probabilities from 0 up to x. “At least” adds probabilities from x up to n, although sometimes it is faster to use a complement. “Between” adds each exact probability in the requested interval. This is where step-by-step output helps: many students know the formula but are unsure which probabilities to add.
Frequently asked questions
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A binomial experiment has a fixed number of trials, each trial has only two outcomes, the probability of success stays the same, and the trials are independent. If any of these conditions fail, a binomial model may not fit the problem. For example, drawing cards without replacement changes probabilities, so a hypergeometric model may be better than a binomial model.
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Success does not always mean something good. It simply means the outcome being counted. If the problem counts defective items, then a defective item is the “success.” If it counts correct answers, then a correct answer is the success. Defining success clearly is the first step before using the formula.
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For P(X ≥ x), add the probabilities for x, x+1, x+2, and so on up to n. In some problems, it is easier to use the complement: P(X ≥ x) = 1 − P(X ≤ x−1). This calculator uses direct addition so the student can see the event clearly.
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The mean is μ = np. It tells you the expected number of successes over many repetitions of the same experiment. If n = 20 and p = 0.30, the expected number of successes is 6. The actual result in one experiment can be different, but 6 is the long-run average.
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Binomial probability counts the number of successes in a fixed number of trials. Geometric probability counts how many trials it takes until the first success occurs. If the problem says “in 10 trials,” think binomial. If it says “until the first success,” think geometric.