poisson probabilities calculator

What is the Poisson distribution?

The Poisson distribution models how many times an event happens in a fixed interval, when events occur independently and at a roughly constant average rate. Typical examples include incoming calls per minute, defects per meter of wire, customer arrivals per hour, or typos per page.

If your process has an expected count of λ per interval, then the number of events X in one interval can often be approximated as Poisson: X ~ Poisson(λ).

Poisson probability formula

For an exact count k:

P(X = k) = e^-λ λ^k / k!

• λ is the mean number of events per interval (also the variance).
• k is a non-negative integer (0, 1, 2, ...).
• e is Euler’s number (~2.71828).

How to use this calculator

1) Enter λ (average rate)

Example: if you expect 3 support tickets per hour on average, then λ = 3 for a 1-hour interval.

2) Choose a probability type

• P(X = k) gives one exact count.
• P(X ≤ k) gives probability up to and including k.
• P(X ≥ k) gives upper-tail probability.
• P(a ≤ X ≤ b) gives probability inside a count range.

3) Enter integer bounds and calculate

The result is shown as a decimal and as a percentage. For convenience, the calculator also reminds you that for Poisson, mean = variance = λ.

Worked examples

Example A: Exact probability

Suppose λ = 3 and you want exactly 2 events. Then: P(X = 2) ≈ 0.2240, or about 22.40%.

Example B: Cumulative probability

With λ = 3, probability of at most 2 events: P(X ≤ 2) ≈ 0.4232, or 42.32%.

Example C: Tail probability

With λ = 3, probability of at least 5 events: P(X ≥ 5) ≈ 0.1847, or 18.47%.

When the Poisson model is appropriate

• Events are counted in a fixed interval (time, area, distance, volume).
• Events are approximately independent.
• The average rate is roughly constant in that interval.
• Two events are unlikely to happen at exactly the same instant.

Common mistakes to avoid

• Using non-integer k: event counts must be whole numbers.
• Mismatched interval: if λ is per hour, do not use it directly for 10 minutes without scaling.
• Ignoring changing rates: if activity changes by time of day, use different λ values for each period.
• Forgetting assumptions: heavy dependence or bursty behavior can break a basic Poisson model.

Poisson vs. Binomial vs. Normal

Use Poisson for counts over intervals with a known average rate. Use Binomial for a fixed number of trials with success/failure outcomes. Use Normal for continuous measurements (or as an approximation for large-count settings).

Practical applications

• Quality control: defects per batch
• Operations: arrivals per minute at a service desk
• IT reliability: failures per day
• Healthcare: incidents per shift
• Traffic and logistics: requests per route window

If you work with queueing, reliability, A/B monitoring, or operations analytics, this Poisson probabilities calculator gives a fast first-pass estimate for event-count risk and planning.