Binomial Distribution Calculator
Calculate binomial probabilities, mean, variance, and cumulative distributions.
Calculate P(X = k) and P(X ≤ k) for Binomial(n, p)
What Is the Binomial Distribution Calculator?
The Binomial Distribution Calculator computes the probability of exactly k successes in n independent trials, where each trial has probability p of success. It also computes the cumulative probability P(X ≤ k), and shows the mean, variance, and standard deviation of the distribution.
Formula
How to Use
Enter n (number of trials), p (probability of success, 0–1), and k (number of successes you're interested in). Click Calculate to see P(X = k), P(X ≤ k), and the distribution statistics.
Example Calculation
Toss a fair coin 10 times. What is the probability of exactly 3 heads? n=10, p=0.5, k=3 C(10,3) = 120 P(X=3) = 120 × (0.5)³ × (0.5)⁷ = 120 × 1/1024 ≈ 11.72% P(X≤3) = P(0)+P(1)+P(2)+P(3) ≈ 17.19% Mean = 10×0.5 = 5, Std Dev = √(10×0.5×0.5) ≈ 1.58
Understanding Binomial Distribution
The binomial distribution is one of the most fundamental probability distributions in statistics. It models any situation with a fixed number of independent trials, each with the same two-outcome probability structure.
The distribution is symmetric when p = 0.5 (like a fair coin), right-skewed when p < 0.5, and left-skewed when p > 0.5. As n increases, it becomes more bell-shaped due to the Central Limit Theorem.
Pascal's triangle, discovered centuries before the formal theory, contains the binomial coefficients C(n,k) as its entries. The connection between combinatorics (counting) and probability (measuring likelihood) is one of the deep unifying themes in mathematics.
Frequently Asked Questions
When does the binomial distribution apply?
Use the binomial distribution when: (1) there are a fixed number n of trials, (2) each trial has only two outcomes (success/failure), (3) trials are independent, and (4) p is constant for each trial.
What is the binomial coefficient C(n,k)?
C(n,k) = n!/(k!(n-k)!) counts the number of ways to choose k successes from n trials. It is also written as "n choose k" or ⁿCₖ.
What is the difference between binomial and Bernoulli?
A Bernoulli distribution is a single trial (n=1) with probability p of success. A binomial distribution is the sum of n independent Bernoulli trials.
When does binomial approximate to normal?
When np ≥ 5 and n(1−p) ≥ 5, the binomial distribution is well approximated by Normal(μ=np, σ²=np(1−p)). This is the Central Limit Theorem in action.
What are real-world examples of binomial distributions?
Quality control (number of defective items in a batch), medical trials (number of patients responding to treatment), survey responses (number agreeing with a statement), and sports (number of free throws made).
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