Binomial Distribution Calculator
Use this online calculator to compute exact and cumulative binomial probabilities. Enter your number of trials, success probability, and the type of probability you want.
What is a binomial distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. If you have n trials and success probability p, then the random variable X (number of successes) follows a binomial distribution.
Common examples include:
- Number of customers who click an ad out of 100 visitors
- Number of defective items in a sample from a production line
- Number of heads in a series of coin flips
- Number of students who pass a quiz in a class
Core formula
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 (the number of ways to choose k successes out of n trials).
Useful summary values
- Mean: E[X] = np
- Variance: Var(X) = np(1 - p)
- Standard deviation: √(np(1 - p))
How to use this online calculator
- Enter the total number of trials n.
- Enter the success probability p (between 0 and 1).
- Select the probability type:
- P(X = k) for exact probability
- P(X ≤ k) for cumulative lower-tail probability
- P(X ≥ k) for upper-tail probability
- P(a ≤ X ≤ b) for a bounded interval
- Enter the needed value(s) and click Calculate.
Practical interpretation tips
1) Match the model assumptions
A binomial model works best when trials are independent and each trial has the same success probability. If probability changes from trial to trial, another model may fit better.
2) Use cumulative probabilities for decision rules
In quality control, risk management, and testing scenarios, cumulative probabilities are often more useful than exact values. For example, “What is the chance of at most 2 defects?” is a classic lower-tail question.
3) Compare expected vs observed results
The expected number of successes is np. If your observed values repeatedly differ a lot from this benchmark, it can signal changing conditions or model misspecification.
Example scenarios
- Marketing: Probability that at least 15 users convert out of 100 visits when conversion rate is 12%.
- Operations: Probability that no more than 1 part is defective in a batch sample of 20 with defect rate 3%.
- Education: Probability that exactly 8 out of 12 students pass when pass probability is 0.7.
- Sports analytics: Probability a player makes at least 4 of 6 free throws with success probability 0.75.
FAQ
Can n be large?
Yes, but very large values can increase computation time for cumulative probabilities. This tool uses numerically stable calculations for most everyday use cases.
What if k is outside 0 to n?
The exact probability is zero in that case. For cumulative forms, results are automatically bounded between 0 and 1.
Can I use decimal k values?
No. The number of successes must be an integer. This calculator expects integer inputs for k, a, and b.