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Poisson Probability Engine

Calculate the exact probability of k events occurring in a fixed interval given an average rate λ. Outputs P(X=k), P(X≤k), and P(X≥k) using the Poisson distribution.

Calculate the precise probability of a specific number of discrete random events occurring over a fixed interval.

Distribution Inputs

Probability Analysis

Exact Match P(X = k)

22.4042%

Cumulative Less P(X < k)

42.3190%
Probability of 0 to 2 events

Cumulative More P(X > k)

35.2768%
Probability of 4+ events
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What is The Poisson Distribution: Modeling Rare Events in Fixed Intervals?

Named after French mathematician Siméon Denis Poisson (1837), the Poisson distribution models the number of independent events occurring in a fixed interval (time, area, or volume) when events happen at a known constant average rate. It is the go-to distribution for rare event modeling: server errors per hour, radioactive decay events per second, typos per page, or traffic accidents per intersection per year.

Mathematical Foundation

Poisson Probability Mass Function
P(X=k)=λke−λk!P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}
P(X=k)P(X = k)= Probability of exactly k events occurring in the interval.
λ\lambda= Average rate parameter — the expected number of events per interval (must be > 0).
kk= The specific number of events you want the probability for (non-negative integer).
ee= Euler's number ≈ 2.71828. The exponential decay term e^(-λ) ensures probabilities sum to 1.

Laws & Principles

  • Three Requirements for Validity: (1) Events must be independent — one event does not affect the probability of another. (2) The average rate λ must be constant throughout the interval. (3) Two events cannot occur at exactly the same instant.
  • Mean = Variance = λ: A unique property of the Poisson distribution is that its mean and variance are both equal to λ. If observed variance is much larger than the mean, the data is overdispersed and a Negative Binomial distribution is more appropriate.
  • Poisson as Binomial Limit: When n is large and p is small (with np = λ), the Binomial distribution converges to the Poisson. This is the 'law of rare events' — 1,000 servers each with a 0.1% failure probability (λ = 1) follow Poisson.

Step-by-Step Example Walkthrough

" A call center receives an average of 4 calls per hour (λ = 4). What is the probability of receiving exactly 6 calls in one hour? "

  1. 1. Identify: λ = 4 calls/hour, k = 6.
  2. 2. Apply formula: P(X=6) = (4⁶ × e⁻⁴) / 6!
  3. 3. Compute: 4⁶ = 4,096. e⁻⁴ = 0.01832. 6! = 720.
  4. 4. P(X=6) = (4,096 × 0.01832) / 720 = 75.07 / 720 = 0.1042.
Final Result: There is a 10.42% probability of receiving exactly 6 calls in one hour. The cumulative P(X ≤ 6) = 88.93%, meaning there is an 89% chance of receiving 6 or fewer calls.

Quick Answer: How does the Poisson Calculator work?

Enter the average rate λ and the target event count k. The calculator applies P(X=k) = λᵏe⁻ᵏ/k! to compute exact, cumulative, and complementary probabilities.

Poisson Formula

P(X = k) = λᵏ × e⁻λ / k! | Mean = Variance = λ

Where λ is the average event rate and k is the target count. The factorial k! grows extremely fast, keeping high-k probabilities small.

Real-World Applications

Server Reliability

If a web server crashes an average of 0.5 times per month (λ = 0.5), the probability of zero crashes in a given month is P(0) = e⁻⁰·⁵ = 60.65%. The probability of 2+ crashes is only 9.02%. This drives SLA (Service Level Agreement) calculations and redundancy planning.

Quality Control

A factory producing circuit boards averages 2 defects per 100 boards (λ = 2). The probability of finding 5+ defects in a batch triggers an investigation. P(X ≥ 5) = 1 - P(X ≤ 4) = 5.27%. This sets the statistical control limits for manufacturing processes.

Poisson Probability Reference (λ = 1 to 5)

k λ = 1 λ = 2 λ = 3 λ = 5
036.79%13.53%4.98%0.67%
136.79%27.07%14.94%3.37%
218.39%27.07%22.40%8.42%
36.13%18.04%22.40%14.04%
50.31%3.61%10.08%17.55%

Poisson Best Practices (Pro Tips)

Do This

  • ✓Check that mean ≈ variance in your data. If your observed variance is much larger than the mean, your data is overdispersed and the Poisson model will underestimate extreme event probabilities.

Avoid This

  • ✗Don't use Poisson when events aren't independent. If one event triggers or prevents another (e.g., earthquake aftershocks), the independence assumption is violated and probabilities will be wrong.

Frequently Asked Questions

When should I use Poisson vs. Binomial?

Use Binomial when you have a fixed number of trials with a known probability per trial. Use Poisson when counting events in a continuous interval (time/space) with a known average rate. As n→∞ and p→0 with np = λ, Binomial converges to Poisson.

Can λ be a decimal?

Yes. λ represents the average rate, which is often non-integer. For example, if a store averages 2.5 customers per minute, λ = 2.5. Only the event count k must be a non-negative integer (0, 1, 2, ...).

What does the e⁻λ term represent?

It is the probability of zero events occurring — and it acts as a normalizing factor. For λ = 5, e⁻⁵ = 0.0067 means only a 0.67% chance of zero events. This term ensures all probabilities P(X=0) + P(X=1) + ... sum to exactly 1.

How do I scale λ for different time intervals?

Multiply λ proportionally to the interval. If λ = 3 per hour, then for a 2-hour window use λ = 6. For 15 minutes, use λ = 0.75. The key is that the rate must remain constant across the interval you choose.

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