Binomial Distribution Calculator
Calculate binomial probability ($P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$), cumulative probabilities ($P(X \le k), P(X \ge k)$), expected mean ($np$), and variance.
Calculate Binomial Probability B(n, p)
Enter number of trials (n), success probability (p), and target successes (k).
Calculation Results
Calculated using binomial distribution probability mass function: P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}
Quick Summary
The Binomial Distribution Calculator evaluates exact probability $P(X = k)$, cumulative probabilities $P(X \le k)$ and $P(X \ge k)$, expected value ($np$), and variance ($np(1-p)$).
Formula Explanation
P(X = k) = \frac{n!}{k!(n - k)!} p^k (1 - p)^{n - k}
E(X) = n \cdot p, \quad \text{Var}(X) = n \cdot p \cdot (1 - p)How It Works
A binomial experiment consists of $n$ identical Bernoulli trials with constant success probability $p$. The Binomial Distribution Calculator evaluates combinations $\binom{n}{k}$, exact probabilities $P(X = k)$, cumulative sums, and distribution mean $E(X) = np$.
Step-by-Step Worked Example
Practical Problem: A fair coin is flipped $n = 10$ times ($p = 0.50$). Calculate probability of getting exactly $k = 5$ heads.
- Step 1: Calculate combinations $\binom{10}{5}$: $\binom{10}{5} = \frac{10!}{5! \, 5!} = \mathbf{252}$.
- Step 2: Calculate $p^k$: $0.50^5 = \mathbf{0.03125}$.
- Step 3: Calculate $(1-p)^{n-k}$: $0.50^{10-5} = 0.50^5 = \mathbf{0.03125}$.
- Step 4: Multiply terms: $P(X = 5) = 252 \times 0.03125 \times 0.03125 = \mathbf{0.24609375\text{ (24.61\%)}}.$
- Step 5: Calculate Expected Mean & Variance: $E(X) = 10 \times 0.5 = \mathbf{5.00}$; $\text{Var}(X) = 10 \times 0.5 \times 0.5 = \mathbf{2.50}$.
Real-World Calculation Examples
Scenario 1: 10 Coin Flips (5 Heads)
Parameters: n = 10, p = 0.50, k = 5
Result: P(X = 5) = 0.2461 (24.61% chance).
Scenario 2: Quality Control Defect Inspection
Parameters: n = 20 parts, p = 0.05 defect rate, k = 0
Result: P(X = 0) = 0.3585 (35.85% zero defect probability).
Scenario 3: Multiple Choice Exam Guessing
Parameters: n = 10 questions, p = 0.25 (4 choices), k = 5 correct
Result: P(X = 5) = 0.0584.
Scenario 4: Sales Call Conversion Rate
Parameters: n = 50 calls, p = 0.10 conversion, k ≥ 5
Result: P(X ≥ 5) = 0.5688.
Key Benefits of Using This Calculator
Exact & Cumulative Probabilities
Calculates $P(X = k)$, $P(X \le k)$, and $P(X \ge k)$ simultaneously.
Expected Mean & Variance
Outputs theoretical mean $E(X) = np$, variance $np(1-p)$, and standard deviation.
Bernoulli Trial Assumptions
Enforces strict Bernoulli trial conditions (independent trials, constant $p$).
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a binomial distribution?
A binomial distribution models the number of successes $k$ in $n$ independent binary trials with constant success probability $p$.
What are the 4 requirements for a binomial experiment?
1. Fixed number of trials $n$. 2. Two binary outcomes (success/failure). 3. Constant probability $p$. 4. Independent trials.
What is formula for binomial mean and variance?
Mean $E(X) = n p$; Variance $\text{Var}(X) = n p (1 - p)$; Standard deviation $\sigma = \sqrt{n p (1 - p)}$.
How do I calculate binomial probability in Excel?
Use formula =BINOM.DIST(k, n, p, FALSE) for exact $P(X=k)$ or =BINOM.DIST(k, n, p, TRUE) for cumulative $P(X \le k)$.
When can binomial distribution be approximated by normal distribution?
When $n p \ge 5$ and $n(1 - p) \ge 5$, the normal distribution $N(np, np(1-p))$ provides a close approximation.
When can binomial distribution be approximated by Poisson distribution?
When $n$ is large ($n \ge 100$) and $p$ is small ($p \le 0.05$), Poisson distribution with $\lambda = np$ approximates binomial.
What is difference between Bernoulli and binomial distribution?
A Bernoulli trial is a single trial ($n = 1$); a binomial distribution is the sum of $n$ Bernoulli trials.
What is combination formula nCr?
$\binom{n}{k} = \frac{n!}{k!(n - k)!}$, representing ways to choose $k$ items from $n$.
Can success probability p be 0 or 1?
Yes — if $p = 0$ or $1$, the distribution becomes deterministic with zero variance.
What is cumulative binomial distribution $P(X \le k)$?
$P(X \le k) = \sum_{i=0}^{k} P(X = i)$, summing probabilities from 0 up to $k$ successes.