Binomial Probability Calculator
Calculate exact and cumulative binomial probabilities using the formula for independent Bernoulli trials.
What is the binomial probability distribution?
The binomial distribution models the number of successes in a fixed number of trials when each trial has only two outcomes: success or failure. It is one of the most practical probability models in statistics, quality control, finance, and data science.
Typical examples include: number of customers who buy, number of heads in coin flips, number of defective units in a batch, or number of emails opened in a campaign.
When you can use a binomial model
- The number of trials n is fixed in advance.
- Each trial has exactly two outcomes (success/failure).
- The probability of success p stays constant across trials.
- Trials are independent of each other.
Binomial probability distribution formula
Exact probability: P(X = k) = C(n, k) × pk × (1 − p)n − k
Combination term: C(n, k) = n! / [k!(n − k)!]
Where:
- X = random variable (number of successes)
- n = number of trials
- k = number of successes of interest
- p = probability of success per trial
Cumulative forms
- At most: P(X ≤ k) = Σ P(X = i), from i = 0 to k
- At least: P(X ≥ k) = Σ P(X = i), from i = k to n
- Between: P(k1 ≤ X ≤ k2) = Σ P(X = i), from i = k1 to k2
How to use this calculator
- Enter the total number of trials n.
- Enter the success probability p as a decimal between 0 and 1.
- Choose whether you want exact, at most, at least, or a range probability.
- Enter k (and k2 for range mode) and click Calculate.
The tool returns the probability in decimal and percentage form, plus the binomial mean, variance, and standard deviation.
Worked examples
Example 1: Exact probability
Suppose 12 customers visit a page, and each has a 30% chance of converting. What is the probability of exactly 4 conversions? Here, n = 12, p = 0.30, k = 4. Use Exact: P(X = k).
Example 2: At least probability
If a quiz has 15 true/false questions and a student guesses each answer (p = 0.5), what is the probability of scoring at least 10 correct? Use At least: P(X ≥ k) with n = 15, p = 0.5, k = 10.
Interpreting results
- Small probabilities are common for very specific outcomes (like exactly k).
- Cumulative probabilities are usually easier to interpret for decisions.
- Compare your result with the expected value E(X) = np to understand whether an outcome is above or below expectation.
Common mistakes to avoid
- Using percentages instead of decimals for p (enter 0.2, not 20).
- Choosing k outside the valid range 0 to n.
- Applying binomial logic when trials are not independent.
- Using this model when p changes from trial to trial.
Quick reference values
- Mean: μ = np
- Variance: σ² = np(1 − p)
- Standard deviation: σ = √[np(1 − p)]
Final thoughts
A binomial probability distribution formula calculator is a fast and accurate way to evaluate success/failure outcomes in real-world scenarios. Whether you are solving homework problems, running A/B tests, or building risk models, this tool helps you move from intuition to precise probability.