Free Binomial Probability Calculator
Use this tool to compute exact and cumulative binomial probabilities for independent trials with a constant success probability.
What is a binomial calculator?
A binomial calculator helps you find the probability of getting a certain number of successes in repeated independent trials. It is based on the binomial distribution, one of the most useful models in statistics, finance, risk analysis, quality control, and A/B testing.
You can use it when each trial has only two outcomes (success/failure), each trial is independent, and the probability of success stays the same.
Binomial distribution formula
The probability of exactly k successes in n trials with success probability p is:
P(X = k) = C(n, k) pk(1-p)n-k
Where C(n, k) (also called “n choose k”) is the number of ways to pick k successes from n trials.
Quick interpretation
- n = number of attempts (coin flips, emails sent, inspected products)
- p = chance of success on each attempt
- k = number of successes you want to evaluate
How to use this online binomial calculator
- Enter n (total trials).
- Enter p as decimal or percent.
- Choose a calculation type:
- P(X = k) exact probability
- P(X ≤ k) at most k successes
- P(X ≥ k) at least k successes
- P(a ≤ X ≤ b) probability in a range
- Click Calculate to see the probability and distribution summary.
Example: email campaign conversion probability
Suppose you send 20 emails and each recipient has a 15% chance of converting. You want the chance of getting exactly 4 conversions.
- n = 20
- p = 0.15
- k = 4
- Type = P(X = k)
The result gives your exact conversion probability. You can switch to P(X ≥ 4) if your goal is “at least 4 conversions.”
When to use binomial vs normal/Poisson
Use binomial when:
- Fixed number of trials
- Two outcomes per trial
- Constant probability of success
- Independent trials
Consider alternatives when:
- Trial count is not fixed (Poisson may fit events over time)
- Outcomes are not binary
- Probability changes from trial to trial
Common mistakes to avoid
- Confusing P(X = k) with P(X ≤ k) or P(X ≥ k).
- Entering percentage incorrectly (e.g., 20 instead of 0.20). This calculator accepts both.
- Using the model when trials are dependent.
- Using non-integer values for n or k.
Helpful output metrics
Along with probability, this calculator shows:
- Mean: μ = np
- Variance: σ² = np(1-p)
- Standard deviation: σ = √(np(1-p))
These are useful for understanding the center and spread of likely outcomes.
Final thoughts
If you need a fast and reliable binomial probability calculator online, this page is built for practical use: exact values, cumulative probabilities, and range probabilities in one place. Save it for statistics homework, experiment design, forecasting, and decision-making.