Binomial Distribution Calculator
Calculate binomial probabilities for a fixed number of independent trials. This calculator computes exact probabilities (P(X = k)), cumulative probabilities (P(X ≤ k)), and range probabilities (P(X < k), P(X > k), P(a ≤ X ≤ b)). Includes a bar chart visualization and step-by-step calculations.
What Is the Binomial Distribution?
The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has exactly two possible outcomes: success or failure. It is one of the most widely used discrete probability distributions in statistics.
A binomial experiment requires:
- A fixed number of trials (n)
- Each trial is independent
- Each trial has only two outcomes: success or failure
- The probability of success (p) is the same for every trial
Binomial Probability Formula
The probability of getting exactly k successes in n trials is given by the probability mass function (PMF):
Where:
- C(n,k) = n! / [k! · (n-k)!] , the binomial coefficient
- p = probability of success on a single trial
- k = number of successes (0, 1, 2, ..., n)
Cumulative Distribution Function (CDF)
The cumulative probability P(X ≤ k) is the sum of all individual probabilities from 0 up to k:
Key Properties
- Expected Value: E(X) = n · p
- Variance: Var(X) = n · p · (1-p)
- Standard Deviation: σ = √[n · p · (1-p)]
Real-World Examples
- Number of heads in 10 coin flips
- Number of free throws made in a game
- Number of defective items in a production batch
- Number of clicks on an online advertisement
- Number of patients who respond to a treatment
Why Use This Binomial Distribution Calculator?
- Multiple Probability Types: Exact, cumulative, less than, greater than, and range probabilities.
- Visual Chart: See the probability mass function with highlighted bars.
- Step-by-Step Breakdown: Understand how the calculation works.
- Complete Statistics: Mean, variance, and standard deviation are automatically computed.
- Free & Private: No registration, no data storage.
❓ Binomial Distribution Calculator FAQ
What is the binomial distribution?
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success.
What is the binomial probability formula?
The probability of exactly k successes in n trials is P(X = k) = C(n,k) · pk · (1-p)n-k, where C(n,k) is the binomial coefficient.
What is the difference between PMF and CDF?
The PMF (Probability Mass Function) gives the probability of exactly k successes. The CDF (Cumulative Distribution Function) gives the probability of k or fewer successes.
What are the parameters of the binomial distribution?
The binomial distribution has two parameters: n (the number of trials) and p (the probability of success on each trial).
What is the expected value of a binomial distribution?
The expected value (mean) is E(X) = n · p. For example, if you flip a coin 10 times (n=10, p=0.5), you expect 5 heads.
What is the variance of a binomial distribution?
The variance is Var(X) = n · p · (1-p). The standard deviation is the square root of the variance.
When should I use the binomial distribution?
Use the binomial distribution when you have a fixed number of independent trials, each with only two outcomes (success/failure), and the probability of success is constant.
What is a Bernoulli trial?
A Bernoulli trial is a single experiment with exactly two possible outcomes: success or failure. A binomial distribution is the sum of n independent Bernoulli trials.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required. All calculations are performed in your browser.