Poisson Probability Calculator

What Is the Poisson Distribution?

The Poisson distribution models the number of events occurring in a fixed interval when events happen independently at a constant average rate.

Parameter:

• λ (lambda) = average rate of events per interval

The question: What's P(X = k), the probability of exactly k events?

The Formula

$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$

Where $e \approx 2.71828$ is Euler's number.

Example: A call center receives 4 calls per hour on average. What's P(exactly 6 calls in the next hour)?

$P(X = 6) = \frac{e^{-4} \times 4^6}{6!} = \frac{0.0183 \times 4096}{720} = 0.104$

About 10.4% chance.

When Does Poisson Apply?

The Poisson distribution requires:

1. Events occur independently: One event doesn't affect others
2. Constant rate: The average rate doesn't change over the interval
3. Proportional to interval: Probability scales with interval length
4. Rare events: Probability of 2+ events in tiny interval is negligible

Classic Examples

• Calls to a call center per hour
• Typos per page
• Cars passing a point per minute
• Radioactive decay events per second
• Server requests per minute
• Goals in a soccer match
• Accidents at an intersection per year

Not Appropriate For

• Events that tend to cluster (use negative binomial)
• Events with varying rates (use non-homogeneous Poisson)
• Events that are not independent

Applications

Queueing Theory

"A bank has 3 customers per hour on average. What's P(exactly 5 customers in an hour)?"

λ = 3, k = 5 P(X = 5) = (e^(-3) × 3^5) / 5! = 0.101

Reliability Engineering

"A machine fails 2 times per year on average. What's P(0 failures next year)?"

λ = 2, k = 0 P(X = 0) = e^(-2) × 2^0 / 0! = 0.135

About 13.5% chance of no failures.

Biology

"A microscope slide has 8 bacteria per cm² on average. What's P(exactly 10 in a given cm²)?"

λ = 8, k = 10 P(X = 10) = (e^(-8) × 8^10) / 10! = 0.099

Sports Analytics

"A team scores 1.5 goals per game on average. What's P(exactly 2 goals)?"

λ = 1.5, k = 2 P(X = 2) = (e^(-1.5) × 1.5²) / 2! = 0.251

Insurance

"Claims arrive at 0.5 per day on average. What's P(3+ claims tomorrow)?"

P(X ≥ 3) = 1 - P(X ≤ 2) = 1 - [P(0) + P(1) + P(2)]

Mean and Variance

A unique property of the Poisson distribution:

$E(X) = \lambda$

$Var(X) = \lambda$

$SD(X) = \sqrt{\lambda}$

Mean equals variance. If observed data has variance much different from the mean, the Poisson model may not fit.

Changing the Interval

If λ is the rate per unit time, for a different interval:

• Rate for 2 hours = 2λ
• Rate for half hour = λ/2
• Rate for any interval t = λt

Example: If 4 calls per hour, then:

• Per 30 minutes: λ = 2
• Per 2 hours: λ = 8

Historical Development

Siméon Denis Poisson (1781-1840) introduced the distribution in 1837, studying the probability of convictions in court cases.

The distribution became famous through Ladislaus Bortkiewicz's 1898 analysis of Prussian cavalry soldiers killed by horse kicks—a classic example of rare events.

A.K. Erlang applied Poisson processes to telephone call modeling in the early 1900s, founding queueing theory.

Relationship to Binomial

The Poisson distribution arises as a limit of the binomial:

If n → ∞, p → 0, and np → λ, then:

Binomial(n, p) → Poisson(λ)

Rule of thumb: When n ≥ 20 and p ≤ 0.05, use Poisson with λ = np.

This explains why Poisson works for rare events: many trials (n), small probability (p), moderate expected count (λ = np).

Cumulative Probabilities

$P(X \leq k) = \sum_{i=0}^{k} P(X = i)$

$P(X \geq k) = 1 - P(X \leq k-1)$

Example: "What's P(at least 3 calls)?"

P(X ≥ 3) = 1 - [P(0) + P(1) + P(2)]

Normal Approximation

For large $\lambda$ (typically $\lambda > 10$):

$X \sim N(\lambda, \sqrt{\lambda})$ approximately

With continuity correction:

$P(X \leq k) \approx \Phi\left(\frac{k + 0.5 - \lambda}{\sqrt{\lambda}}\right)$

Overdispersion

When variance > mean, data shows "overdispersion"—more variability than Poisson predicts. Causes include:

• Clustering of events
• Heterogeneous rates across subjects
• Correlation between events

Solutions: negative binomial regression, quasi-Poisson models.

Poisson Regression

When modeling count data with predictors:

$\log(\lambda) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \ldots$

Coefficients exponentiate to rate ratios:

• $e^{\beta_1}$ = multiplicative effect on rate per unit increase in $X_1$

Related Calculations

• Poisson Rate Calculator — Estimate λ from data
• Binomial Probability Calculator — Discrete alternative
• Normal Probability Calculator — Approximation for large λ
• Confidence Interval for Mean — Interval for λ

Summary

The Poisson distribution models the count of independent events occurring at a constant average rate λ. From call centers to radioactive decay to sports scores, Poisson probabilities answer questions about event counts. The distribution's key property—mean equals variance—provides a check for model fit. As a limit of the binomial for rare events, the Poisson distribution is fundamental to probability theory and widely applied in queueing, reliability, and count data analysis.