Binomial Distribution Calculator

Calculate binomial probabilities: P(X = k), P(X ≤ k), P(X ≥ k). Find mean, variance, and standard deviation for binomial experiments with n trials and success probability p.

📊 Statistics📐 P(X=k) = C(n,k) × pᵏ × (1-p)^(n-k)🔢 Math
Number of trials n
Probability of success p
Number of successes k
Probability type
Probability type
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Binomial Distribution Calculator—P(X=k) = C(n,k) × pᵏ × (1-p)^(n-k)probability

Step-by-Step Examples

Example 1
Coin Flips

Fair coin, n=10 flips. P(exactly 3 heads)?

  • P(X=3) = C(10,3) × 0.5^3 × 0.5^7
  • = 120 × 0.125 × 0.0078125 = 0.1172
  • Mean = 5, SD = 1.58
✓ P(X=3) = 11.72%
Example 2
Quality Control

5% defect rate, n=20. P(at least 2 defective)?

  • P(X>=2) = 1 - P(X=0) - P(X=1)
  • P(X=0) = 0.95^20 = 0.3585
  • P(X=1) = 20×0.05×0.95^19 = 0.3774
  • P(X>=2) = 0.2641
✓ P(X>=2) = 26.41%
Example 3
Drug Response

p=0.7, n=8. P(all 8 respond)?

  • P(X=8) = 0.7^8 = 0.0576
  • Only 5.76% chance all 8 respond
  • Expected responders: 8×0.7 = 5.6
✓ P(X=8) = 5.76%

Real-World Applications

🧪
Clinical Trials
Binary trial outcomes modeled as binomial for power analysis.
💼
Quality Control
Number of defective items in a batch follows binomial distribution.
🏀
Sports
Free throw success rate: how many will a player make from n attempts?
📊
Survey Analysis
Yes/no responses in a survey of n people with probability p.

Common Mistakes to Avoid

⚠️
Forgetting C(n,k) in formula

P(X=k) includes C(n,k) = n!/(k!(n-k)!) the number of ways to arrange k successes in n trials.

⚠️
Large n: use approximation

For n>100 with small p, use Poisson (lambda=np). For moderate p, use normal approximation.

⚠️
Checking independence assumption

Binomial requires independent trials. If drawing without replacement from a small population, use hypergeometric distribution instead.

Frequently Asked Questions

What is binomial distribution? ▾
Models successes in n independent trials each with probability p. Classic: coin flips. Mean=np, Variance=np(1-p).
When to use normal approximation? ▾
When np>5 and n(1-p)>5, use Normal(np, sqrt(np(1-p))). Add continuity correction of ±0.5.
What is C(n,k)? ▾
The binomial coefficient n!/(k!(n-k)!): number of ways to choose k items from n. C(10,3)=120.
What is the negative binomial? ▾
Counts trials needed for k successes instead of successes in n trials.
What is cumulative probability? ▾
P(X<=k) = sum of P(X=0) through P(X=k). Important for finding probabilities of at-most or at-least events.

Related Math Calculators

📊 Normal Distribution📊 P-Value🔢 Combinations📊 Poisson Distribution📊 Expected Value

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

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