Binomial & Poisson Calculator — n,p,k; PMF, CDF, Mean & Variance

Understanding the Binomial and Poisson Distributions

The binomial distribution models the number of “successes” in a fixed number of independent trials. You specify n (number of trials) and p (success probability for each trial). If \( X \sim \mathrm{Bin}(n,p) \), then the probability of exactly k successes is \( P(X=k)=\binom{n}{k}p^k(1-p)^{\,n-k} \). The expected value and variance are \( \mathbb{E}[X]=np \) and \( \mathrm{Var}(X)=np(1-p) \). Typical uses include pass/fail testing, survey responses, and quality control where the number of attempts is known in advance and each attempt has the same chance of success.

The Poisson distribution models the number of events occurring in a fixed interval of time, distance, or area when events happen independently at a constant average rate \( \lambda \) (lambda). If \( X \sim \mathrm{Pois}(\lambda) \), then \( P(X=k)=e^{-\lambda}\lambda^k/k! \), with \( \mathbb{E}[X]=\lambda \) and \( \mathrm{Var}(X)=\lambda \). It is widely used for arrivals (calls per minute, patients per hour, defects per metre), and other “count per interval” scenarios across the US, UK, EU, India, and beyond.

When to Choose Binomial vs Poisson

• Use Binomial when the number of opportunities is fixed (n known), each trial is independent, and the success probability is constant (p).
• Use Poisson when you count events over a continuous interval with a stable rate \( \lambda \) and independence between events.
• Connection: When n is large and p is small with \( \lambda=np \) moderate, Poisson provides a handy approximation to Binomial.

Reading the Graph and Tail Probabilities

The bar chart shows the probability mass function (PMF) \(P(X=k)\) for each integer \(k\). Our summary also reports the cumulative distribution function (CDF) \(P(X\le k)\) and the right tail \(P(X\ge k)\). Left tails are useful for questions like “no more than k”, while right tails answer “at least k”. In education settings (GCSE/A-Level, AP/College) and in industry (manufacturing QA/QC, call centres, hospitals), tail probabilities help with decision thresholds, alarms, and service-level guarantees.

Approximations & Practical Tips

• Normal approximation (Binomial): If \(np(1-p)\) is reasonably large (rule of thumb: \(\ge 10\)), \(X\) is often approximated by \( \mathcal{N}(np,\;np(1-p)) \). For better accuracy on discrete counts, apply a continuity correction (e.g., use \(k+0.5\)).
• Normal approximation (Poisson): For large \( \lambda \), \(X\) may be approximated by \( \mathcal{N}(\lambda,\;\lambda) \) with a continuity correction.
• Rate vs probability: In Poisson problems, ensure the rate \( \lambda \) matches the interval you are analysing (per hour, per day, per km). Rescale if needed.

Common Pitfalls

• Non-independence: If trials or events influence each other, both models can mislead.
• Changing probabilities/rates: If p or \( \lambda \) varies across trials or time, consider more advanced models (e.g., negative binomial for over-dispersion).
• Parameter bounds: Binomial requires \(0 \le p \le 1\) and integer \(0 \le k \le n\). Poisson requires \( \lambda \ge 0 \) and integer \( k \ge 0 \).

Note: This calculator supports coursework and professional work across the US, UK (modelling/modelling), EU, India, Australia, and more—use it to estimate defect rates, queue arrivals, pass rates, and service targets with clear PMF/CDF visuals.