Poisson Probability
P(X = k) = (λᵏ × e^(-λ)) / k!
The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when events happen independently at a constant average rate. It is commonly used for rare events and count data.
Variables
The probability of observing exactly k events
The average number of events per interval
The specific number of occurrences to calculate the probability for
The mathematical constant approximately equal to 2.71828
Example Calculation
Scenario
A hospital emergency room receives an average of 4 patients per hour. What is the probability of receiving exactly 6 patients in a given hour?
Given Data
Calculation
P(X = 6) = (4⁶ × e^(-4)) / 6! = (4096 × 0.01832) / 720 = 75.07 / 720
Result
P(X = 6) = 0.1042 or about 10.4%
Interpretation
There is approximately a 10.4% chance of exactly 6 patients arriving in any given hour. While this is above the average of 4, the Poisson distribution has a right skew, so moderately higher counts are reasonably probable.
When to Use This Formula
- ✓Modeling the number of events in a fixed interval when events occur independently at a constant rate
- ✓Approximating binomial probabilities when n is large and p is small (rare events)
- ✓Analyzing count data such as defects per unit, calls per hour, or accidents per month
- ✓Queuing theory and operations research applications
Common Mistakes
- ✗Using the Poisson distribution when events are not independent or the rate varies over time
- ✗Forgetting that λ must be expressed for the same interval as the question asks about
- ✗Confusing P(X = k) with P(X ≤ k) and not summing when cumulative probabilities are needed
- ✗Applying the Poisson model when the variance is much larger or smaller than the mean (overdispersion or underdispersion)
Calculate This Formula Instantly
Snap a photo of any problem and get step-by-step solutions.
Download StatsIQFAQs
Common questions about this formula
Use the Poisson distribution when counting events in a continuous interval (time, area, volume) rather than in a fixed number of trials. The Poisson can also approximate the binomial when n is large (n > 20), p is small (p < 0.05), and λ = np is moderate. In such cases, the Poisson is computationally simpler.
Scale λ proportionally. If the average is 4 events per hour and you want the probability for a 30-minute window, use λ = 2 (half of 4). For a 2-hour window, use λ = 8. The Poisson distribution adjusts naturally because λ represents the expected count for whatever interval you specify.