Binomial Probability Calculator
Use this tool to compute exact, cumulative, or range probabilities for a binomial random variable.
What this find binomial probability calculator does
This calculator helps you quickly compute probabilities from a binomial distribution, which is one of the most common probability models in statistics. You can find the probability of getting an exact number of successes, up to a number of successes, at least a number of successes, or a range of successes.
Typical use cases include quiz outcomes, manufacturing defects, A/B testing conversions, and any repeated yes/no process where each trial has the same success chance.
Binomial probability formula
If X ~ Binomial(n, p), then:
where C(n, k) = n! / (k!(n-k)!)
Here, n is the number of trials, p is the probability of success on each trial, and k is the number of successes you are evaluating.
How to use the calculator
- Enter n: the number of independent trials.
- Enter p: probability of success per trial (between 0 and 1).
- Select your probability type: exact, at most, at least, or between.
- Enter the required value(s) for x (and b if using a range).
- Click Calculate to see the probability in decimal and percent form.
Worked examples
Example 1: Exactly x successes
A fair coin is flipped 10 times. What is the probability of exactly 5 heads? Set n=10, p=0.5, and choose P(X = x) with x=5.
Example 2: At least x successes
A marketing email has a 20% click rate. If 30 users are contacted, what is the chance of at least 8 clicks? Set n=30, p=0.2, choose P(X ≥ x), and enter x=8.
Example 3: Between two values
A factory defect rate is 3%. Out of 120 products, what is the chance of between 1 and 5 defects? Set n=120, p=0.03, and choose P(a ≤ X ≤ b) with a=1 and b=5.
When binomial modeling is appropriate
You should use a binomial model when all of the following are true:
- There are a fixed number of trials (n is known).
- Each trial has only two outcomes (success/failure).
- Trials are independent.
- The success probability p stays constant across trials.
Common mistakes to avoid
- Using percentages like 20 instead of decimals like 0.20.
- Entering non-integer values for trial counts or success counts.
- Using binomial assumptions when trials are not independent.
- Confusing at most (≤) with at least (≥).
Quick reference
- Mean: E(X) = np
- Variance: Var(X) = np(1-p)
- Standard deviation: √(np(1-p))
If you are studying statistics or running real-world experiments, this find binomial probability calculator can save time and reduce calculation errors. Try different values to build intuition about how changing n and p shifts the distribution.