Binomial Probability Calculator
Compute probabilities for a binomial random variable X ~ Bin(n, p).
What is a binomial calculator?
A binomial calculator helps you find probabilities for events that have exactly two outcomes on each trial: success or failure. Examples include flipping a coin (heads/tails), testing a manufactured part (pass/fail), or tracking whether users click an ad (click/no click).
Instead of calculating large combinations by hand, this tool computes common binomial probabilities in seconds: exact outcomes, cumulative outcomes, and bounded ranges.
The binomial model in one formula
Exact probability
For a random variable X ~ Bin(n, p), the probability of exactly k successes is:
P(X = k) = C(n, k) · pk · (1-p)n-k
- n = number of independent trials
- p = success probability on each trial
- k = number of successes you want
- C(n, k) = number of combinations of k successes among n trials
Cumulative probability
Cumulative values are sums of exact probabilities. For example: P(X ≤ k) adds probabilities from 0 through k, while P(X ≥ k) adds probabilities from k through n.
How to use this calculator binomial tool
- Enter n (total trials).
- Enter p (success chance per trial from 0 to 1).
- Select the probability type (exact, at most, at least, or range).
- Enter k, or a and b for a range.
- Click Calculate to view probability and percentage.
The result panel also displays the mean, variance, and standard deviation for the same binomial setup.
Practical examples
1) Coin flip scenario
Suppose you flip a fair coin 10 times. With n = 10 and p = 0.5, you can find the chance of exactly 6 heads, at most 4 heads, or between 3 and 7 heads.
2) Quality control
If 2% of parts are defective, set p = 0.02 and n to your sample size. You can quickly estimate the chance of seeing zero defects, one defect, or multiple defects.
3) Marketing conversion
If a campaign converts at p = 0.12 and you expect 25 visitors, use the calculator to estimate how likely it is to get at least 5 conversions.
When should you use a binomial distribution?
Use binomial methods when all of the following are true:
- You have a fixed number of trials.
- Each trial has two outcomes (success/failure).
- The success probability p stays constant across trials.
- Trials are independent (one result does not affect another).
Common mistakes to avoid
- Using percentages like 40 instead of decimals like 0.40 for p.
- Entering non-integers for n or k.
- Using binomial when outcomes are not independent.
- Forgetting that p must be between 0 and 1 inclusive.
Quick FAQ
Is this a normal distribution calculator?
No. This is specifically for binomial probabilities, though large-sample binomial models are sometimes approximated by normal distributions.
Can I calculate “at least” probabilities?
Yes. Choose P(X ≥ k) and enter your minimum success count.
What does the percentage mean?
It is the same probability expressed on a 0–100 scale. For example, 0.237 equals 23.7%.