Binomial vs Poisson Distribution: When to Use Each

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Three required assumptions: (1) Fixed number of trials, n. (2) Each trial has only two possible outcomes (success or failure). (3) Independent trials with constant probability p. Probability mass function. P(X = k) = C(n,k) × p^k × (1-p)^(n-k), where C(n,k) is the binomial coefficient "n choose k." Mean = np. Variance = np(1-p). Classic examples. (1) Number of heads in 10 coin flips (n=10, p=0.5). (2) Number of defective items in a sample of 50 (n=50, p=defect rate). (3) Number of customers who purchase in a sample of 100 visitors (n=100, p=conversion rate). All three: fixed sample size, binary outcome, constant probability per trial. When binomial fits poorly. (1) If the probability varies trial-to-trial (heterogeneous outcomes). (2) If trials are not independent (clustered customer behavior, repeated measurements on same units). (3) If the number of trials is not fixed in advance.

The Poisson distribution models the number of events occurring in a fixed interval (time, space, volume) when events occur independently at a constant average rate. Three required assumptions: (1) Events occur independently. (2) Average rate (lambda) is constant. (3) Probability of two events at exactly the same instant is zero. Probability mass function. P(X = k) = (lambda^k × e^(-lambda)) / k!, where lambda is the average number of events per interval. Mean = lambda. Variance = lambda (these are equal — a defining feature of the Poisson distribution). Classic examples. (1) Number of customer arrivals per hour at a retail store. (2) Number of typos per page in a manuscript. (3) Number of phone calls received per minute at a help desk. (4) Number of radioactive decays per second from a sample. All four: counting events in a fixed interval, constant average rate, independent occurrences. When Poisson fits poorly. (1) If the rate varies over time (overdispersion). (2) If events cluster (positive correlation). (3) If variance >> mean in the data (overdispersion). Common sign of misfit: in real data, variance often exceeds the Poisson mean, suggesting a negative binomial distribution (which has a separate variance parameter) may fit better.

The two distributions are related but applied to different problems. | Feature | Binomial | Poisson | |---|---|---| | Models | Number of successes in n trials | Number of events in interval | | Required parameter(s) | n (trials), p (success prob) | lambda (rate) | | Mean | np | lambda | | Variance | np(1-p) | lambda | | Mean = Variance? | No (unless p=0.5) | Yes (defining feature) | | Domain | 0, 1, 2, ..., n | 0, 1, 2, ... (unbounded) | | Sum of independent | Binomial(n1+n2, p) if same p | Poisson(lambda1+lambda2) | | Common in | Quality control, conversion testing | Arrivals, defects, counts | Key conceptual differences. Binomial has a hard cap at n (cannot have more successes than trials). Poisson is unbounded — in principle 1,000,000 events could occur, just with vanishingly small probability. Binomial requires knowing both n and p separately; Poisson collapses these into a single rate parameter.

When n is large and p is small (with np remaining moderate), the binomial distribution approaches the Poisson distribution. Rule of thumb: when n > 20 and p < 0.05 (or n > 100 and p < 0.1), Poisson approximation works well with lambda = np. Proof intuition. As n grows and p shrinks (with np = lambda held constant), the binomial PMF converges to the Poisson PMF. This is why Poisson is sometimes called "the distribution of rare events." Practical use. (1) Computing rare-event probabilities. Defect rate of 0.001 with sample size 10,000 — exact binomial computation involves C(10,000, k) which is computationally heavy. Poisson approximation with lambda = 10 gives the same answer with much simpler math. (2) Manufacturing QA. (3) Counting rare adverse events in clinical trials. The relationship also informs which to use when both could apply. If trials are clearly enumerated (1000 widgets, each can be defective), binomial is the natural choice. If events are arrivals or counts (defects in a continuous strip of fabric), Poisson is natural. When trials are enumerated AND many AND p is small, the choice is mathematically equivalent — use whichever is easier to compute.

Snap a photo of any count data and StatsIQ identifies which distribution fits best (binomial, Poisson, negative binomial, geometric, others). The app computes probabilities under each, runs goodness-of-fit tests, and flags overdispersion that would suggest a different distribution. For study design, StatsIQ produces simulation-based power analyses for both binomial and Poisson outcomes. For exam prep, the app generates problems at varying difficulty levels covering all common discrete distributions. This content is for educational purposes only.