Binomial Distribution Calculator & Bernoulli Histogram
Calculate exact and cumulative binomial probabilities B(n; p) with step-by-step solutions. Features interactive histogram plotting, Gaussian normal curve overlay, sigma intervals, and trainer.
⚙️Verteilungsparameter n und p festlegen
Passe Versuchszahl n und Trefferwahrscheinlichkeit p flexibel an.
📝Ausführlicher Rechenweg & Formelherleitung
Bei $n = 20$ Versuchen und Trefferwahrscheinlichkeit $p = 0,50$ ($q = 1 - p = 0,50$):
Die Formel für genau $k$ Treffer lautet $P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$:
Da $\sigma = 2,24 \le 3$, ist die Verteilung zu schmal für eine stetige Glockenkurven-Approximation.
Historical Origin & The Challenge Solved
Why was this mathematics developed, where does it come from, and what real-world problem did it solve?
1. Quality Control in Coin Minting & Gambler Runs
Royal mints needed to ensure minted gold coins met strict purity tolerances. Authorities had no mathematical model to calculate the probability of finding exactly k defective coins in a batch of 100 samples.
2. Bernoulli Trials & The Binomial Probability Formula
In "Ars Conjectandi", Bernoulli proved the formula P(X=k) = (n choose k) * p^k * (1-p)^(n-k) and established the Law of Large Numbers: observed frequencies converge to theoretical probabilities over large samples.
3. Industrial Quality Assurance, Polling & A/B Testing
Website A/B testing, semiconductor chip manufacturing yield metrics, and national election exit polls rely directly on the binomial distribution.
Praxisnutzen & Reale Anwendungsfälle
Warum lernen wir das? So wird diese Mathematik in Wirtschaft, Technik, Wissenschaft und Alltag praktisch angewendet:
Lot Inspection & Defect Rates
Manufacturers verify defect rates and acceptance thresholds using random sample batches.
Testing 100 components with 2% defect rate for at most 3 failures.
Voter Surveys & Confidence Intervals
Statisticians calculate margins of error for political approval ratings.
Estimating voting proportions from a sample of 1,000 citizens with 95% confidence.
Clinical Drug Trials
Evaluating recovery rates and treatment efficacy in randomized patient groups.
Probability of a new therapy curing at least 80 out of 100 patients.
Mathematical Foundations: Binomial Distribution & Bernoulli Trials
A Bernoulli trial is a random experiment with exactly two outcomes (success with probability p, failure with probability q = 1 - p). Repeating this trial n times independently forms a Bernoulli chain of length n.
The random variable X denotes the number of successes. The probability of exactly k successes is given by the Bernoulli formula: P(X = k) = (n choose k) · p^k · (1-p)^(n-k).
Key metrics are expected value μ = n · p, variance Var(X) = n · p · (1-p), and standard deviation σ = √(n · p · (1-p)). If σ > 3 (Laplace condition), the distribution can be approximated by a continuous Gaussian normal distribution.
Frequently Asked Questions (FAQ) about Binomial Distribution
When is a process binomially distributed?
When each trial has only 2 outcomes, the success probability p remains constant (drawing with replacement), and all trials are independent.
How do you calculate at least k successes P(X ≥ k)?
Using the complement rule: P(X ≥ k) = 1 - P(X ≤ k-1). For at least 1 success, use P(X ≥ 1) = 1 - (1-p)^n.
What are the sigma intervals?
They denote symmetric ranges around μ: The 1σ interval contains ≈68.3% of the distribution, 2σ contains ≈95.4%, and 3σ contains ≈99.7%.