Poisson Probability Calculator
Compute exact, cumulative and tail Poisson probabilities from the mean rate λ and the event count k. The calculator shows the PMF, the CDF, the Poisson process timeline and every algebraic step — all live as you drag the sliders.
λ is both the mean and the variance
For a Poisson distribution, E[X] = λ and Var(X) = λ. That means the standard deviation is √λ, so a larger rate produces both a larger mean and a wider spread — but the relative spread σ/μ = 1/√λ shrinks as λ grows. This is why Poisson counts with large λ look almost normal, while small-λ counts are strongly right-skewed.
The factor 1/k! makes the PMF sum to 1
The probability that exactly k events occur is λk e−λ / k!. The e−λ factor is the same for every k — it is chosen precisely so that Σk≥0 P(X = k) = 1. The 1/k! factor reflects the fact that the order of the k events does not matter: two arrivals in a minute is one outcome, not two.
Small λ → rare events, high skew
When λ < 1, P(X = 0) = e−λ dominates and the distribution is extremely right-skewed: most intervals see zero events. When λ ≈ 1 the distribution flattens. When λ > 10 the distribution becomes almost symmetric and the skewness 1/√λ drops below 0.32 — the normal approximation becomes reasonable.
The interval must be fixed in advance
Poisson counts require a fixed interval — one minute, one page, one hour, one year. Doubling the interval doubles λ and completely changes the distribution. Similarly, events must be independent and occur at a constant rate; clustered events (rush hour, contagion) violate the model and require a negative binomial or a non-homogeneous Poisson process instead.
How to Use This Calculator
The Poisson distribution gives the probability of observing exactly k events in a fixed interval when events occur independently at an average rate λ.
1. Set the mean rate λ
Type a value or drag the slider. λ is the expected number of events in your interval. If you know the rate per unit time and your interval is different, multiply or divide accordingly before entering λ.
2. Set the event count k
k must be a non-negative integer. Use the preset buttons for common values, or drag the slider for a continuous sweep. The chart updates live so you can see the PMF move.
3. Choose what to solve for
Pick P(X = k) for the exact probability, P(X ≤ k) for the cumulative probability of at most k events, or P(X ≥ k) for the tail probability of at least k events. The result card and diagram both update to reflect the choice.
4. Read the result and verdict
The result card shows the probability to six decimal places plus the other two forms (exact, cumulative, tail) for comparison. A colour-coded verdict classifies the probability as very likely, likely, unlikely or rare.
5. Check the diagram and charts
The PMF chart shows the full distribution with the highlighted region shaded according to your selected mode. The diagram adds the mean line, the current k marker and a chip row showing P(X = 0) through P(X = 10).
6. Switch to Advanced for depth
Advanced mode adds a cumulative distribution step chart, a sensitivity panel showing how the probability reacts to ±10% changes in λ and ±1 changes in k, and a full companion statistics table including mean, variance, mode and skewness.
What This Tool Offers
Built around the classic Poisson formula P(X = k) = (λk · e−λ) / k! — the probability of exactly k events in a fixed interval.
Exact, cumulative & tail
Three modes in one tool: P(X = k), P(X ≤ k) and P(X ≥ k). The result card shows all three so you can compare them at a glance.
Live PMF bar chart
The full probability mass function drawn as bars, with the highlighted region shaded according to your selected mode and the mean λ marked as a vertical line.
Cumulative step chart
An advanced-mode view of the CDF as a step function, with the current k highlighted and the distribution's median marked.
Distribution shape diagram
A large annotated PMF visualization with the mean λ marked, the current k highlighted, and a chip row showing P(X = 0) through P(X = 10).
Sensitivity panel
See how the probability changes when λ moves by ±10% or when k moves by ±1 — the elasticity of the result to each input.
Reference tables
Poisson vs Binomial vs Normal comparison, typical λ values by domain, a grouped-rate calculator, and a full variable reference table.
Reference Tables & Extra Tools
A rate-scaling helper plus lookup tables for formulas and typical λ values.
Rate scaling helper
Poisson Formulas at a Glance
For a Poisson-distributed random variable X with rate λ > 0.
| Quantity | Formula | Example (λ = 3, k = 2) |
|---|---|---|
| PMF | P(X = k) = λk e−λ / k! | 0.224042 |
| CDF | P(X ≤ k) = Σi=0..k λi e−λ / i! | 0.423190 |
| Tail (≥ k) | P(X ≥ k) = 1 − P(X ≤ k−1) | 0.800852 |
| Mean | μ = λ | 3 |
| Variance | σ² = λ | 3 |
| Standard deviation | σ = √λ | 1.732051 |
| Mode | ⌊λ⌋ (and ⌊λ⌋−1 if λ is an integer) | 3 |
| Skewness | 1 / √λ | 0.577350 |
| Excess kurtosis | 1 / λ | 0.333333 |
| P(X = 0) | e−λ | 0.049787 |
Poisson vs Binomial vs Normal
Choose the right count distribution for your problem.
| Distribution | Models | Parameters | Mean / Variance | Use when |
|---|---|---|---|---|
| Poisson | Counts in a fixed interval | λ (rate) | λ / λ | Events occur at a constant average rate; no fixed number of trials |
| Binomial | Successes in n trials | n, p | np / np(1−p) | Fixed number of independent trials with constant success probability |
| Normal | Continuous sums and averages | μ, σ | μ / σ² | Large samples; use as approximation when λ > 20 for Poisson |
| Negative binomial | Over-dispersed counts | r, p | rp/(1−p) / rp/(1−p)² | Variance exceeds the mean (clustered events) |
Typical λ Values by Domain
Representative rates — the appropriate λ depends entirely on your chosen interval.
| Domain | Event | Interval | Typical λ |
|---|---|---|---|
| Call centre | Incoming calls | 1 minute | 2 – 15 |
| Web server | Requests | 1 second | 50 – 5,000 |
| Manufacturing | Defects per batch | 1 batch of 1,000 | 0.5 – 5 |
| Epidemiology | New cases | 1 day in a city | 0.1 – 100 |
| Astronomy | Photons from a source | 1 second | 10 – 1,000 |
| Insurance | Claims per policy-year | 1 year | 0.01 – 0.5 |
| Genomics | Mutations per gene | 1 gene | 0.001 – 0.1 |
| Queueing | Arrivals at a till | 1 hour | 5 – 30 |
Variable Reference
| Symbol | Meaning | Typical range | Notes |
|---|---|---|---|
| λ | Mean rate (events per interval) | 0.0001 – 500 | Both mean and variance of the distribution |
| k | Event count | 0 – 500 | Non-negative integer |
| e | Euler's number | ≈ 2.71828 | Base of the natural logarithm |
| k! | Factorial of k | 1 – ∞ | Product 1 × 2 × … × k; 0! = 1 |
| μ | Mean of the distribution | = λ | Same as the rate |
| σ² | Variance of the distribution | = λ | Equidispersion: variance equals mean |
| σ | Standard deviation | = √λ | Same units as k |
What Does P(X = k) = (λk e−λ) / k! Measure?
The Poisson distribution models the number of times an event occurs in a fixed interval of time, space or volume, given that events occur independently at a constant average rate λ. It is the natural distribution for count data with no upper bound: the number of calls arriving at a call centre in a minute, the number of defects in a manufactured batch, the number of radioactive decays in a second, the number of mutations in a gene. The formula P(X = k) = λk e−λ / k! gives the probability that exactly k events occur in that interval.
Two ingredients make the Poisson distribution special. First, the events are independent — one arrival does not make another more or less likely in the next instant. Second, the rate λ is constant over the interval. Under these conditions, the probability of exactly k events depends only on λ, not on the length of the interval in any other way. The mean and variance are both equal to λ, which is why the Poisson distribution is called equidispersed; when the variance exceeds the mean, a negative binomial or a mixed Poisson model is more appropriate.
Poisson Probability Formula
The defining formula is:
The cumulative distribution is the sum of the PMF up to k:
The tail probability of at least k events is the complement:
For numerical stability with large λ and k, the calculator evaluates the log-PMF first:
Poisson Variables
| Symbol | Meaning | Notes |
|---|---|---|
| λ | Mean rate parameter | Also equals the variance; must be positive |
| k | Number of events | Non-negative integer; the value whose probability you want |
| e | Euler's number ≈ 2.71828 | Base of the natural logarithm |
| k! | Factorial of k | k! = k × (k−1) × … × 1; 0! = 1 by convention |
| P(X = k) | Probability of exactly k events | Sum over all k equals 1 |
| P(X ≤ k) | Cumulative probability | Monotone increasing in k, from 0 to 1 |
| P(X ≥ k) | Tail probability | 1 − P(X ≤ k−1); useful for "at least" questions |
How Each Part Affects the Probability
The λk factor grows rapidly with k for large λ, which is why high-rate distributions have their mass concentrated around λ. The e−λ factor is a constant normaliser that shrinks everything for large λ. The 1/k! factor shrinks the tails for large k faster than λk grows, which is why probabilities of very large counts (much greater than λ) fall off faster than exponentially. The combined effect produces the characteristic hump shape: the PMF rises until k ≈ λ, then falls rapidly.
Because λ is both the mean and the variance, the relative spread σ/μ = 1/√λ shrinks as λ grows. For λ = 1 the distribution is extremely skewed (skewness 1.0), for λ = 10 the skewness is 0.316 and the distribution is roughly bell-shaped, and for λ = 100 the skewness is 0.1 and the distribution is visually indistinguishable from a normal curve. This is why the normal approximation is often used for Poisson counts when λ > 20, with a continuity correction.
Poisson Calculation Examples
λ = 3, k = 2
P(X = 2) = (3² · e⁻³) / 2! = (9 × 0.049787) / 2 = 0.224042
λ = 3, k = 2
P(X ≤ 2) = P(0) + P(1) + P(2) = 0.049787 + 0.149361 + 0.224042 = 0.423190
λ = 3, k = 2
P(X ≥ 2) = 1 − P(X ≤ 1) = 1 − 0.199148 = 0.800852
A call centre receives an average of 3 calls per minute. What is the probability of 5 calls in 2 minutes?
λ = 3 × 2 = 6
P(X = 5) = (6⁵ · e⁻⁶) / 5! = (7776 × 0.002479) / 120 = 0.160623
Frequently Asked Questions
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that events occur independently at a constant average rate λ. It is defined for non-negative integers k = 0, 1, 2, … and has the probability mass function P(X = k) = λk e−λ / k!. Its mean and variance are both equal to λ, which makes it a natural model for count data with no upper bound.
Use Poisson when three conditions hold: (1) you are counting events in a fixed interval, (2) the events occur independently of each other, and (3) the average rate λ is constant over the interval. Classic examples: calls arriving at a call centre, defects in a manufactured batch, radioactive decays in a second, mutations in a gene, arrivals at a queue. If the events cluster (rush hour, contagion), the variance exceeds the mean and a negative binomial or mixed Poisson model fits better. If the number of trials is fixed and bounded, use the binomial distribution instead.
λ (lambda) is the mean rate parameter — the expected number of events in the interval of interest. If a call centre receives an average of 4 calls per minute, then for a one-minute interval λ = 4; for a two-minute interval λ = 8; for a 30-second interval λ = 2. λ is also the variance of the distribution, which is a special property of the Poisson: the mean and variance are always equal. This is why the standard deviation is simply √λ.
The binomial distribution models the number of successes in a fixed number of independent trials n with success probability p. The Poisson distribution models the number of events in a fixed interval when there is no fixed number of trials — the count is unbounded. In fact, the Poisson is the limiting case of the binomial as n → ∞ and p → 0 with np = λ fixed. So if you have a very large number of trials each with a very small success probability, the binomial is well approximated by a Poisson with λ = np. Rule of thumb: use Poisson when n > 20 and p < 0.05.
No. The Poisson distribution is discrete and defined only for non-negative integers k = 0, 1, 2, …. The factorial k! is only defined for non-negative integers (and generalised to the gamma function for non-integers, but the Poisson PMF is not used that way). If you have a question about "at least 2.5 events", interpret it as "at least 3 events" — the integer ceiling. This calculator rounds k to the nearest integer on input to keep the calculation well-defined.
For large λ the Poisson distribution becomes nearly symmetric and bell-shaped. The skewness is 1/√λ, so for λ = 100 the skewness is 0.1 and the distribution is visually indistinguishable from a normal curve with mean λ and variance λ. The normal approximation with continuity correction is often used in practice when λ > 20. This calculator evaluates the exact Poisson PMF via logarithms, so it remains accurate even for large λ and k without numerical overflow.
The factor e−λ is a normalising constant. Without it, the sum Σk≥0 λk / k! would equal eλ rather than 1, so multiplying by e−λ makes the total probability sum to exactly 1. It is the same constant for every k, so it does not affect the relative shape of the distribution — it simply scales all probabilities down as λ grows. For small λ (say λ = 0.5), e−λ is close to 1 and the distribution is concentrated at k = 0; for large λ (say λ = 20), e−λ is tiny but λk grows enormously, so the products balance out around k ≈ λ.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| P(X=k) | Probability of exactly k events | |
| λ | Average rate of events | |
| k | Number of events observed |
What it means
The Poisson distribution models the number of events occurring in a fixed interval of time or space, given the average number of events λ. It assumes events are independent and occur at a constant rate. It is used in queuing theory, insurance claim modelling, traffic flow, and epidemiology (rare disease counts). The formula uses the exponential and factorial. The mean and variance both equal λ. Understanding the Poisson distribution is fundamental for modelling count data and for analysing rare events. It is also used in reliability (number of failures) and in biology (number of mutations).
Worked example
Poisson Probability – Two Examples
Real‑World| Parameter | Value |
|---|---|
| λ | 3 |
| k | 2 |
| Parameter | Value |
|---|---|
| λ | 5 |
| k | 5 |
Common mistakes
- Poisson distribution: Models the number of events in a fixed interval of time or space.
- Parameter λ: The average number of events – must be >0.
- k: Non‑negative integer (0,1,2,…).
- e: Euler’s number (~2.718) – use correct value.
- Independence: Assumes events occur independently with a constant rate.
Applications
The Poisson probability formula gives the probability of exactly k events occurring in a fixed interval of time or space, given a known average rate λ. It models rare events and is used in reliability (number of failures), quality control (defects per unit), and queueing theory (arrivals). Engineers use it to predict the number of defects in manufactured goods, to design spare parts inventory, and to assess the probability of extreme events. In biology, it models mutation rates. By using the Poisson distribution, professionals can quantify the likelihood of various counts, set service levels, and plan capacity. Its simplicity and wide applicability make it a staple in statistical practice.
- Quality control – number of defects per product or per batch
- Reliability engineering – number of failures over time
- Queueing theory – number of arrivals or service completions
- Inventory management – demand during lead time
- Biology and ecology – organism counts in a given area
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