Poisson Probability
Probability of k events in a fixed interval given average rate λ.
Formula
Variables
Example
Rare-event modeling like call arrivals
How it works
This formula gives the exact probability of observing k events when events occur randomly and independently at a constant average rate λ over a fixed amount of time, space, or area. It comes from the Poisson process, where the count of events in equal-sized intervals follows a pattern that depends only on the mean rate; the factor e^{-λ} ensures all probabilities add to 1, and k! accounts for the number of indistinguishable orderings of k arrivals. The shape changes with λ: small λ concentrates mass near 0, while larger λ shifts the likely counts upward.
When to use it
Use it when you are counting how many times a rare or randomly scattered event happens in a fixed interval, and the average rate is known or estimated. Typical examples include defects per meter of material, radioactive decays in a time window, typos per page, or bacteria colonies in a small sample area.
Worked example, step by step
Suppose a factory averages λ = 2.4 defective chips per batch, and we want the probability of exactly k = 3 defects in one batch. Step 1: Write the formula: P(X=3) = (λ^k e^{-λ}) / k! Step 2: Substitute values: P(X=3) = (2.4^3 e^{-2.4}) / 3! Step 3: Compute each part: 2.4^3 = 13.824, 3! = 6, and e^{-2.4} ≈ 0.0907 Step 4: Multiply numerator: 13.824 × 0.0907 ≈ 1.2545 Step 5: Divide by 6: 1.2545 / 6 ≈ 0.2091 Final answer: P(X=3) ≈ 0.209, so there is about a 20.9% chance of exactly 3 defective chips in a batch.
Common mistakes to avoid
- Using the formula for event rates that change over time instead of a constant average rate
- Forgetting that k must be a whole number and that k! is factorial, not k times λ
- Mixing up the average rate λ with the event count k or plugging in the wrong time interval
Did You Know?
The Poisson distribution approximates the binomial for rare events.
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