The Poisson distribution models counts of independent events at a steady average rate, such as calls per hour or defects per metre. Enter λ and k for exact and cumulative probabilities.
Enter the average rate λ.
Enter k.
Read the probabilities.
Exactly k: P(X = k) = e^(−λ) λᵏ ÷ k!
The Poisson distribution gives the probability of seeing a certain number of events in a fixed interval of time or space when events happen independently at a steady average rate. The only input it needs is that average rate, λ. This calculator returns the probability of exactly k events, at most k and at least k.
Typical examples are calls reaching a helpline per hour, accidents at a junction per month, typing errors per page, defects per metre of cloth, customers arriving at a bank counter in 10 minutes, and radioactive decays per second. Operations managers use it to plan staffing, insurers to price rare events, and students meet it in Class 12 and college statistics as the distribution of rare events.
1. Fix the interval and find λ, the average number of events in an interval of exactly that size.
2. If your rate is for a different interval, scale it: 3 per hour becomes 1.5 per 30 minutes.
3. Exactly k: P(X = k) = e^(−λ) λᵏ ÷ k!.
4. At most k: add P(X = i) for i = 0, 1, …, k.
5. At least k: 1 − P(X ≤ k − 1), which is the same as 1 − P(X ≤ k) + P(X = k).
6. Remember that the mean and the variance both equal λ.
Split the interval into n tiny pieces, each with a small probability p of containing one event, and let np = λ. The count of events is then binomial. As n grows and p shrinks with λ fixed, the binomial formula C(n, k) pᵏ (1 − p)ⁿ⁻ᵏ turns into e^(−λ) λᵏ ÷ k!. The factor (1 − λ/n)ⁿ approaches e^(−λ), and C(n, k) pᵏ approaches λᵏ ÷ k!. This is why Poisson approximates the binomial well when n is large and p is small, and why it is called the law of rare events.
Three conditions must hold. Events occur one at a time. They are independent, so one event does not make another more or less likely. And the average rate is constant across the interval. Real data can break these. Hospital admissions rise during an outbreak, and traffic accidents cluster at rush hour. A quick diagnostic is that for Poisson data the variance should roughly equal the mean. If the variance is much larger, called overdispersion, the Poisson model will understate the chance of extreme counts.
For large k, both λᵏ and k! overflow ordinary calculators long before their ratio becomes a problem. The calculator avoids this by working with logarithms: it computes −λ + k ln λ − ln(k!) and then takes the exponential, using a log-gamma function for ln(k!). The special case λ = 0 is handled separately, giving probability 1 for zero events and 0 otherwise. The distribution is skewed right for small λ and becomes more symmetric and bell-shaped as λ grows, approaching a normal curve with mean λ and SD √λ.
A bank branch in Jaipur receives an average of 4.5 written customer complaints per working day. The branch manager wants to know the chance of a quiet day with exactly 2 complaints, and of 2 or fewer.
Poisson formula: P(X = k) = e^(−λ) λᵏ ÷ k! = e^(−4.5) × 4.5^2 ÷ 2! = 0.112479
At most k: P(X ≤ 2) = Σ P(X = i), i = 0…2 = 0.173578
At least k: 1 − P(X ≤ 2) + P(X = 2) = 0.938901
Mean and variance: Both equal λ = 4.5
Answer: P(X = 2) 0.112479; P(X ≤ 2) 0.173578; P(X ≥ 2) 0.938901
Using a rate for one interval with a question about another, such as a per-hour λ for a 15-minute window.
Treating 'at least k' as 1 − P(X ≤ k), which leaves out exactly k.
Applying Poisson to events that cluster or depend on each other.
Using Poisson when there is a small fixed number of trials; the binomial is the right model there.
Entering the count of events observed in one period as λ instead of the long-run average.
Staffing call centres, bank counters and hospital emergency departments.
Estimating the probability of rare accidents or insurance claims.
Quality control of defects per unit length or area of cloth, sheet or road surface.
Counting radioactive decays and photon arrivals in physics labs.
Approximating binomial probabilities for large n and small p.
How is λ chosen?
It is the average number of events in the same size interval you are asking about.
When does Poisson approximate binomial?
When n ≥ 20 and p ≤ 0.05, using λ = np.