Poisson Distribution Calculator
Calculate Poisson probabilities for events occurring in a fixed interval.
Quick Examples
Poisson Distribution
The Poisson distribution models the probability of a given number of events occurring in a fixed interval, given a known average rate. It is the distribution of counts without a fixed number of trials: calls per hour, emails per minute, defects per unit, arrivals per day. The single parameter λ is both the mean and the variance.
Probability Mass Function
Key Formulas
Cumulative probabilities add the individual masses: P(X ≤ k) = P(X=0) + P(X=1) + … + P(X=k).
Applications
| Application | Example |
|---|---|
| Call Centers | Calls per hour |
| Quality Control | Defects per batch |
| Traffic | Cars at intersection per minute |
| Biology | Mutations per DNA segment |
Conditions for Poisson
- Events occur independently
- Rate is constant
- Two events cannot occur at exactly the same instant
- Events are rare relative to the interval
Worked Example 1: Help Desk Calls
A help desk receives an average of λ = 4 calls per hour. What is the probability of exactly 6 calls in one hour?
About a 10.4% chance. The cumulative probability of 6 or fewer calls is P(X ≤ 6) ≈ 0.8893, so the chance of more than 6 calls is 1 − 0.8893 = 0.1107.
Worked Example 2: Cumulative Probability
The number of snow days per month in a cold climate is Poisson with a mean of λ = 3. What is the probability of exactly 4 snow days in a month?
About a 16.8% chance. Note the right skew: with λ = 3, outcomes above the mean (4, 5, 6, …) remain quite possible, while negative counts are impossible — the distribution has a floor at zero but no ceiling. Skewness equals 1/√λ, so the distribution becomes more symmetric as λ grows.
Scaling λ to Your Interval
The rate λ must match the interval in your question. If an expert typist makes on average 2 errors every 5 pages, the rate for a 15-page stretch is three times as large:
The probability of at most 5 errors in 15 pages is then P(X ≤ 5) ≈ 0.4457, and the chance of exactly 5 errors is P(X = 5) ≈ 0.1606. Always rescale λ before computing — a rate per hour cannot answer a question about a 30-minute window without halving.
Poisson vs Binomial: Which One?
| Feature | Poisson | Binomial |
|---|---|---|
| What it counts | Events in a fixed interval | Successes in a fixed number of trials |
| Maximum count | No upper limit | Cannot exceed n |
| Parameters | λ (average rate) | n (trials), p (success probability) |
| Mean and variance | Both equal λ | np and np(1−p) |
| Example | Defects per batch | Defective items among 10 inspected |
Poisson as an Approximation to the Binomial
When n is large and p is small (roughly n ≥ 100 and np ≤ 10), the binomial is well approximated by a Poisson distribution with λ = np. Rare defects can be modeled either way; the Poisson form is simpler when n is huge.
Limitations
- Independence: if events cluster — calls arriving in bursts, aftershocks following an earthquake — the model underestimates extremes.
- Constant rate: rush hours, seasons, and machine wear change λ. Split the interval and use a different λ per segment when the rate clearly varies.
- Discrete counts only: k must be a whole number (0, 1, 2, …). The Poisson distribution never produces fractional counts or negative ones.
Common Mistakes
- Forgetting to rescale λ: a rate of 4 calls per hour is λ = 2 for a half-hour question, not 4.
- Confusing λ with k: λ is the average (can be any positive number, e.g. 2.5); k is the observed count (a whole number). They play different roles in the formula.
- Treating the mean as the most likely outcome: P(X = λ) is rarely the maximum probability — the distribution is right-skewed, with skewness 1/√λ.