Compute exact and cumulative binomial probabilities — P(X = k), P(X ≤ k), and P(X ≥ k) — for n independent trials with success probability p, plus the expected number of successes.
Results
Calculated
P(X = k)
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Exact probability of exactly k successes
P(X ≤ k)
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Cumulative probability of k or fewer successes
P(X ≥ k)
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Probability of k or more successes
Mean (expected)
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Expected successes, μ = n × p
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How to use this calculator
This calculator finds binomial probabilities: the chance of getting exactly, at most, or at least a given number of successes across a fixed number of independent trials, each with the same success probability. Enter the number of trials (n), the probability of success (p) as a percentage, and the number of successes (k) you want to evaluate, then click Calculate. Click Clear to reset all fields and start a new calculation.
The formula
The binomial probability mass function is P(X = k) = C(n, k) × pk × (1 − p)n − k, where C(n, k) = n! / (k! (n − k)!) counts the number of ways to arrange k successes among n trials. This applies whenever a trial has exactly two outcomes (success/failure), the trials are independent, and the success probability p stays constant across all n trials — for example, counting heads in 10 coin flips, or defective items in a batch with a fixed defect rate.
Understanding the inputs
Number of trials (n) is the total count of independent attempts (1 to 1000). Probability of success (p) is the chance of success on any single trial, entered as a percentage from 0 to 100. Number of successes (k) is the specific count you want the probability for, and must be a whole number from 0 to n.
Interpreting the results
P(X = k) is the exact probability of landing on precisely k successes. P(X ≤ k) sums that probability with every smaller count, giving the chance of k or fewer successes. P(X ≥ k) is the complementary upper-tail probability of k or more successes — note P(X ≤ k) and P(X ≥ k) overlap at exactly k, so they do not sum to 1 unless you use P(X ≤ k−1) instead. The mean, μ = n × p, is the long-run average number of successes if you repeated the n trials many times.
Frequently Asked Questions
What is the binomial probability formula?
The binomial probability mass function is P(X = k) = C(n, k) × pk × (1 − p)n − k, where n is the number of independent trials, p is the probability of success on a single trial, k is the number of successes, and C(n, k) = n! / (k! (n − k)!) is the number of ways to choose k successes from n trials.
What conditions must be met to use the binomial distribution?
Four conditions must hold: a fixed number of trials n, each trial is independent of the others, each trial has only two possible outcomes (success or failure), and the probability of success p stays constant across every trial. Coin flips and yes/no survey questions with a fixed sample size are classic examples.
What is the difference between P(X = k) and P(X ≤ k)?
P(X = k) is the exact probability of getting precisely k successes. P(X ≤ k), the cumulative probability, is the probability of getting k or fewer successes and is the sum of P(X = i) for every i from 0 to k. P(X ≥ k) is the complementary upper-tail probability of getting k or more successes.
What are the mean and standard deviation of a binomial distribution?
The mean (expected number of successes) is μ = n × p. The standard deviation is σ = √(n × p × (1 − p)). For example, with n = 10 trials and p = 0.5, the mean is 5 successes and the standard deviation is about 1.58.
Practical Guide for Binomial Distribution Calculator - Probability Calculator (Binomial) - Math Calculator
Binomial Distribution Calculator - Probability Calculator (Binomial) - Math Calculator is most useful when the inputs reflect the situation you are actually planning around, not a best-case estimate. Treat the result as a decision aid: it gives you a structured way to compare assumptions, spot outliers, and decide what to verify next. For Math work, the most important review lens is formula choice, units, rounding, weighting, and the exact meaning of each input.
Start with a baseline run using values you can defend. Then change one assumption at a time and watch which output moves the most. If one input dominates the result, spend your verification time there first. If several inputs have similar influence, use a conservative scenario and an optimistic scenario to create a practical range instead of relying on a single exact number.
Before acting on the result, verify the result with a manual calculation or a second method when the output affects grades, budgets, or engineering work. This is especially important when the calculator supports a purchase, project plan, performance target, or operational decision. The calculator can make the math consistent, but the quality of the conclusion still depends on current data, clear units, and assumptions that match your real constraints.
When the output looks surprising, slow down and inspect each input in order. A small change in one high-leverage field can move the final number more than several low-leverage fields combined. For Binomial Distribution Calculator - Probability Calculator (Binomial) - Math Calculator, that means you should first confirm the value with the greatest scale, then confirm the value with the greatest uncertainty, then rerun the calculator with conservative and optimistic assumptions. This sequence turns the calculator from a single answer into a practical decision range.
Review Checklist
Confirm every input uses the unit and time period requested by the calculator.
Run a low, expected, and high scenario so the answer has a useful range.
Check whether rounding or a missing decimal place changes the decision.
Update the calculation after each new value is known or whenever the formula structure changes.