Online Binomial Distribution Calculator
Calculate exact and cumulative binomial probabilities in seconds. Enter your values, choose the probability type, and click Calculate.
Tip: x values should be integers. The calculator uses the binomial PMF and CDF formulas.
What is a binomial distribution?
A binomial distribution models the number of successes in a fixed number of independent trials when each trial has the same probability of success. If you have n trials and each trial has success probability p, then the random variable X (number of successes) follows a binomial distribution: X ~ Binomial(n, p).
This is one of the most practical probability models in statistics, data science, finance, quality control, and exam analytics. Whether you are asking “What is the chance of exactly 3 wins in 10 games?” or “How likely are at least 8 correct answers out of 12 questions?”, a binomial distribution calculator online gives quick, accurate answers.
Binomial formula used by this calculator
The exact probability of getting exactly k successes is:
P(X = k) = C(n, k) · pk · (1 - p)n-k
where C(n, k) is the combination term (“n choose k”).
- Exactly: P(X = x)
- At most: P(X ≤ x) = sum of P(X = k) for k = 0 to x
- At least: P(X ≥ x) = sum of P(X = k) for k = x to n
- Between: P(x₁ ≤ X ≤ x₂) = sum of P(X = k) for k = x₁ to x₂
How to use this binomial distribution calculator online
Step 1: Enter your distribution parameters
Input the total number of trials (n) and probability of success on each trial (p).
For example, if each trial has a 30% chance of success, enter 0.3 for p.
Step 2: Choose the probability type
Select whether you need exact, cumulative lower tail, cumulative upper tail, or interval probability.
Step 3: Enter x (or x₁ and x₂)
For exact/at most/at least, enter one integer. For “between,” enter two integer bounds.
Step 4: Click calculate
The calculator returns the probability and summary statistics (mean, variance, standard deviation) for the selected binomial model.
Real-world examples
- Education: Probability a student answers exactly 15 out of 20 multiple-choice questions correctly by skill or guessing model.
- Marketing: Number of conversions from 100 ad clicks when conversion probability is 0.04.
- Manufacturing: Defective items in a production batch where each item has fixed defect risk.
- Sports: Wins over a season with constant per-game win probability.
- Medical studies: Response counts in repeated yes/no outcomes.
Quick interpretation guide
If the resulting value is 0.12, that means the event has a 12% chance under your assumptions. Very small probabilities can indicate rare events; larger cumulative probabilities can help with threshold decisions and risk planning.
Useful properties of Binomial(n, p)
- Mean: n × p
- Variance: n × p × (1 − p)
- Standard deviation: √(n × p × (1 − p))
Common mistakes to avoid
- Using percentages as whole numbers (enter 0.25, not 25).
- Using non-integer x values for success counts.
- Applying binomial logic when trials are not independent.
- Using changing p values across trials (binomial requires constant p).
FAQ
Can I use this for large n?
Yes. The script uses a numerically stable log-gamma approach for combinations and works well for typical practical inputs.
Is this the same as normal distribution?
No. Binomial is discrete (counts), while normal is continuous. For large n, a normal approximation may sometimes be used, but this calculator computes binomial probabilities directly.
Does “between” include endpoints?
Yes. The “between” mode calculates P(x₁ ≤ X ≤ x₂), which includes both bounds.