Binomial Probability Calculator
Compute exact and cumulative binomial probabilities instantly.
What this online binomial calculator does
This tool computes probabilities for a binomial random variable using the classic binomial model. In plain language, it answers questions like: “If I repeat an experiment n times and each trial has success probability p, what is the chance of getting exactly (or at most/at least) a certain number of successes?”
- Exact probability: P(X = k)
- Cumulative probability: P(X ≤ k)
- Right-tail probability: P(X ≥ k)
- Range probability: P(k1 ≤ X ≤ k2)
Binomial distribution refresher
A binomial distribution models the total number of successes in a fixed number of independent Bernoulli trials. A Bernoulli trial has only two outcomes: success or failure.
Use the binomial model when all are true
- You run a fixed number of trials, n.
- Each trial is independent of the others.
- Each trial has only two outcomes (success/failure).
- The probability of success, p, is constant across trials.
Formula used
where C(n, k) = n! / (k!(n-k)!)
The calculator also builds cumulative results from these probabilities, so you can get “at most,” “at least,” and interval probabilities without doing multiple manual calculations.
How to use the calculator
- Enter n: total number of trials.
- Enter p: success probability per trial (0 to 1).
- Select the probability type you want.
- Enter k (and k2 for a range query).
- Click Calculate.
Practical examples
Example 1: Exactly k successes
Suppose you send 15 sales emails and historical response probability is 0.2. If you want the chance of getting exactly 4 responses, enter n=15, p=0.2, and choose P(X = k) with k=4.
Example 2: At most k failures/successes
You quality-check 30 items, and each item has a 5% defect rate. To find the chance of at most 1 defect, use n=30, p=0.05, calculation type P(X ≤ k), and k=1.
Example 3: A probability range
In 50 independent signups with a conversion probability of 0.12, you might ask: “What is the chance of getting between 4 and 9 conversions?” Use P(k1 ≤ X ≤ k2), with k1=4 and k2=9.
Understanding the output
Along with the requested probability, the tool returns the distribution’s key moments:
- Mean: μ = np
- Variance: σ² = np(1-p)
- Standard deviation: σ = √(np(1-p))
These values help you interpret center and spread. For planning and forecasting, they’re often just as useful as the single probability value.
Common mistakes to avoid
- Using percentages instead of decimals for p (write 0.35, not 35).
- Applying binomial assumptions when p changes from trial to trial.
- Using binomial for dependent events.
- Confusing “at most” (≤) with “at least” (≥).
Where binomial probability is used
- A/B testing and conversion analysis
- Reliability and defect-rate checks
- Medical trial outcomes (success/failure endpoints)
- Survey response modeling
- Risk estimation in operations and finance
Final note
If your experiment follows the four binomial conditions, this calculator gives fast, reliable answers for exact and cumulative binomial probability queries. Keep inputs clean, match the right query type, and you can move from raw assumptions to actionable probability in seconds.