How to use it
- Enter the average rate λ for your interval.
- Enter k, the count you want the probability for.
- Read the exact and cumulative probabilities. The chart highlights k.
How it works
The probabilities are built from P(0) = e^(−λ) using P(i) = P(i − 1) · λ / i, in logarithms so a large λ does not underflow. Each tail is summed directly, and the upper tail is summed far enough out that what is left over is below double precision.
Worked example
λ = 4 and k = 2: P(X = 2) = e⁻⁴ · 16 / 2 = 8e⁻⁴ ≈ 0.1465251. The cumulative P(X ≤ 2) = e⁻⁴ (1 + 4 + 8) = 13e⁻⁴ ≈ 0.2381033.
Tips
- The model assumes events are independent and the rate is constant.
- “At least one event” is P(X ≥ 1) = 1 − e^(−λ).