Introduction
A binomial distribution tells you the probability of getting a certain number of successes in a fixed number of trials, where each trial has only two outcomes: success or failure. For example, if you flip a coin 10 times, what are the chances you get exactly 7 heads? That's the kind of question a binomial distribution answers.
This Binomial Distribution Calculator makes it easy to find exact and cumulative probabilities without doing the math by hand. Enter the number of trials (n), the probability of success (p), and the number of successes (x) you want to evaluate. You can calculate an exact probability like P(X = x), or switch to cumulative range mode to find the probability that the number of successes falls between two values. The calculator instantly shows results for P(X = x), P(X ≤ x), P(X ≥ x), P(X < x), and P(X > x), along with summary statistics like the mean, variance, standard deviation, skewness, and kurtosis. It also displays the step-by-step formula, a full probability distribution table, and an interactive bar chart so you can see the shape of the distribution at a glance.
How to use our Binomial Distribution Calculator
Enter the details of your binomial experiment below, and this calculator will give you exact and cumulative probabilities, summary statistics, a distribution chart, and a full probability table.
Number of Trials (n): Enter the total number of independent trials in your experiment. This must be a whole number that is zero or greater. For example, if you flip a coin 10 times, enter 10.
Probability of Success (p): Enter the chance that a single trial results in a success. This value must be between 0 and 1. For example, a fair coin has a probability of 0.5 for landing on heads. If you need to convert a percentage to a decimal, simply divide by 100.
Calculation Mode: Choose how you want to calculate your probability. Select the "Exact Value" option, P(X = x), to find the probability of getting exactly a certain number of successes. Select the "Cumulative Range" option, P(x₁ ≤ X ≤ x₂), to find the probability that the number of successes falls within a range you define.
Number of Successes (x): This field appears when you choose Exact Value mode. Enter the specific number of successes you want to find the probability for. It must be a whole number between 0 and n. For example, enter 5 if you want to know the chance of getting exactly 5 heads in 10 flips.
Min Successes (x₁) and Max Successes (x₂): These fields appear when you choose Cumulative Range mode. Enter the lowest and highest number of successes for your range. Both must be whole numbers, and x₁ must be less than or equal to x₂, with both values between 0 and n. For example, set x₁ to 3 and x₂ to 7 to find the probability of getting between 3 and 7 successes.
What Is a Binomial Distribution?
A binomial distribution describes the probability of getting a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. Think of it like flipping a coin 10 times and asking, "What are the chances I get exactly 6 heads?" Every flip is one trial, heads is a success, and the probability stays the same each time.
The Three Key Inputs
To use the binomial distribution, you need three pieces of information:
- n (number of trials): How many times you repeat the experiment. For example, rolling a die 20 times means n = 20.
- p (probability of success): The chance of success on any single trial, expressed as a decimal between 0 and 1. A fair coin has p = 0.5, while rolling a six on a standard die has p = 1/6 ≈ 0.1667.
- x (number of successes): The specific count of successful outcomes you want to find the probability for.
The Binomial Formula
The probability of getting exactly x successes in n trials is calculated using this formula:
P(X = x) = C(n, x) · px · (1 − p)n − x
Here, C(n, x) is the binomial coefficient, sometimes written as "n choose x." It counts how many different ways you can arrange x successes among n trials. The term px accounts for the probability of the successes, and (1 − p)n − x accounts for the probability of the failures.
Exact vs. Cumulative Probabilities
Sometimes you need more than just the exact probability. That's where cumulative probabilities come in:
- P(X = x): The chance of getting exactly x successes.
- P(X ≤ x): The chance of getting x or fewer successes. This is the cumulative distribution function (CDF).
- P(X ≥ x): The chance of getting x or more successes.
- P(x₁ ≤ X ≤ x₂): The chance of getting between x₁ and x₂ successes, inclusive. This is a cumulative range.
Summary Statistics Explained
The binomial distribution also has useful summary numbers that describe its overall shape:
- Mean (μ = n · p): The average number of successes you would expect over many repetitions of the experiment.
- Variance (σ² = n · p · (1 − p)): A measure of how spread out the results are from the mean.
- Standard Deviation (σ): The square root of the variance, which puts the spread in the same units as the data.
- Skewness: Tells you whether the distribution leans left or right. When p = 0.5, the distribution is perfectly symmetric and skewness equals zero.
- Kurtosis (excess): Describes how peaked or flat the distribution is compared to a normal distribution.
When to Use the Binomial Distribution
The binomial distribution applies when four conditions are met: (1) you have a fixed number of trials, (2) each trial is independent of the others, (3) each trial has only two outcomes (success or failure), and (4) the probability of success stays the same on every trial. Common real-world examples include quality control testing (how many defective items in a batch), medical studies (how many patients respond to a treatment), survey analysis (how many people answer "yes"), and games of chance like dice or card draws with replacement.