How to use
Use this calculator to estimate how many independent events occur in a fixed interval: for example, arrivals during ten minutes or flaws along a fixed length of material. Enter the average count λ for that exact interval and the whole count k you want to examine. λ is an average, so it does not have to be an integer. Choose exactly k, at most k or more than k, then press Calculate. The highlighted bars and main result refer to that same event.
Understand the probabilities
The probability of exactly k events is P(X=k)=exp(−λ)λᵏ/k!. The cumulative probability P(X≤k) adds all counts from zero through k. P(X>k) includes k+1 and every larger count; it is the complement of the cumulative probability. The mean and variance both equal λ. These are properties of the model, not a guarantee about the next observed interval.
Worked examples
With λ=2 and k=3, the exact probability is about 18.04%, the probability of at most three events is 85.71%, and the probability of more than three is 14.29%. For an average of two arrivals per ten minutes, the last result describes four or more arrivals in ten minutes.
With λ=1 and k=0, the probability of no events is 36.79%. The probability of at least one is 63.21%. To model a period twice as long at the same steady rate, double λ first: an average of two per ten minutes becomes four per twenty minutes. Do not leave λ unchanged while changing the count interval.
Uses and limits
This model assumes independent occurrences at a constant average rate. Clusters, changing demand and a hard upper limit can make it unsuitable. Supported inputs are 0≤λ≤10,000 and integer 0≤k≤100,000. At λ=0, zero events are certain. Blank values, negative counts and fractional counts produce an error beside the field. Decimal point and decimal comma are accepted, but thousands separators are not.
Frequently asked questions
Why are some counts missing from the picture?
The chart shows 31 consecutive integer counts around the mean, or starts at zero for a small mean. It never groups several counts into one bar. The caption gives the probability outside this window and the selected boundary, even when that boundary is off screen. All reported probabilities use the full distribution.
Does a tall bar mean an area probability?
For this discrete distribution, a bar’s height is the probability of one count. Add the selected heights to obtain the selected probability, including any selected counts outside the window. Bar widths are only for readability.
Can a very small probability appear as zero?
Small positive values use scientific notation. Values below floating-point resolution are marked as less than 1E−300 when the event remains possible. Other displayed numbers are rounded to at most two decimal places; calculations retain their internal precision. A value displayed as 100% can be rounded, rather than logically certain.
How do I save or change a result?
Copy result copies the confirmed inputs, event and numeric results as text. Editing any input or selection clears the old result and picture and disables Copy until you calculate again. Clear empties numeric inputs. The tool does not fit λ from a dataset, save a history or export a chart.