Poisson Probability Calculator

Formula Used

How to Use This Calculator

1. Enter the event rate λ in events per unit.
2. Enter the exposure t for your time window or scale.
3. Choose the probability mode that matches your question.
4. Provide k and optionally k₂.
5. Press Calculate to see results above the form.

Common physics uses include radioactive decay counts, photon arrivals, detector hits, and rare-event statistics.

Poisson Probability Guide

Poisson Counts in Experimental Physics

Many detectors produce integer counts: radioactive decays, photon arrivals, ion hits, and rare trigger events. When events are independent and occur at a roughly constant average rate, the Poisson model is a practical first description for the count distribution within a fixed exposure window.

Key Inputs: Rate and Exposure

This calculator separates the physical rate λ (events per unit) from the exposure t (time, area, length, or any scale that linearly accumulates opportunity). Their product μ = λt is the expected count. For example, λ = 20 counts/s and t = 0.10 s gives μ = 2 expected counts.

Interpreting μ, Mean, and Variance

A defining feature of the Poisson distribution is that the mean and variance are both μ. This matters for uncertainty: the standard deviation is √μ, so relative noise scales as √μ/μ = 1/√μ. Doubling μ improves relative precision by about 1/√2, a common rule of thumb in counting statistics.

Exact Probability for a Single Count

Use Exact: P(X = k) when you want the probability of observing one specific count. If μ = 2, the probabilities are P(0) = e⁻² ≈ 0.1353 and P(2) = e⁻² 2²/2! ≈ 0.2707. Exact probabilities are useful for validating simulations and checking discrete outcomes.

Cumulative and Tail Probabilities

Engineering and physics decisions often depend on thresholds. P(X ≤ k) answers “How likely is a low count?” while P(X ≥ k) answers “How surprising is a high count?” If μ = 6 and you observe k = 12, the tail probability P(X ≥ 12) can quantify how unusual that spike is under the assumed rate.

Interval Probabilities for Acceptance Windows

Interval mode computes P(k ≤ X ≤ k₂), useful for defining pass bands in quality control, coincidence windows, or expected background ranges. For μ = 4, the probability of falling between 2 and 6 counts summarizes the “typical” region more directly than a single exact value.

Practical Data Examples in the Lab

In photon counting, a calibrated source might yield λ = 500 s⁻¹. With t = 2 ms, μ = 1.0, so P(X = 0) ≈ 0.3679 and P(X ≥ 1) ≈ 0.6321. In radiation monitoring, λ = 0.2 s⁻¹ over t = 60 s gives μ = 12, where √μ ≈ 3.46 sets a natural scale for fluctuations.

Limits, Assumptions, and Good Practice

The model assumes independent events and a stable average rate across the exposure. Dead time, pileup, drifting sources, or correlated bursts can break these assumptions. When μ becomes very large, the Poisson distribution approaches a normal distribution with mean μ and variance μ, which this calculator may use for stable cumulative estimates.

FAQs

1) What does λ represent in a physics experiment?

λ is the average event rate per unit exposure, such as decays per second or hits per square centimeter. It should represent a stable average during the measurement window.

2) What is μ and why is it important?

μ = λt is the expected number of events in the exposure window. In a Poisson model, μ is also the mean and the variance, so it controls both the average count and the noise level.

3) When should I use P(X ≥ k) instead of P(X = k)?

Use P(X ≥ k) to quantify how surprising a high observed count is under your assumed rate, such as when testing alarms, rare spikes, or exceedance thresholds.

4) Can this calculator handle large k or large μ?

Yes. Exact probabilities use a log-factorial approach to reduce overflow. For very large inputs, cumulative results may switch to a normal approximation for numerical stability.

5) What does P(X = 0) tell me physically?

P(X = 0) is the probability of observing no events during the exposure. It is common in low-rate experiments, dark counts, background checks, and “no-hit” detector validations.

6) How do I interpret the standard deviation √μ?

√μ is the typical fluctuation size in counts around the mean. If μ = 100, then √μ = 10, so counts near 90–110 are common under the model.

7) What assumptions should I verify before trusting results?

Check independence of events and a roughly constant rate across the exposure. If there is dead time, burstiness, or drift, a Poisson model may under- or over-estimate probabilities.

Example Data Table

Values are rounded for display. Your results depend on your exact inputs.