Skip to content
Calcumatix

Poisson Distribution Calculator

Find the chance of an exact, at-most, or at-least event count from a mean rate and a whole-number target in statistics problems.

Result

14.65%

For a Poisson distribution with mean 4, P(X = 2) is 0.1465 (14.65%)

Quick Answer

A Poisson distribution calculator evaluates count probabilities in a fixed interval. For lambda = 4 and k = 2, P(X = 2) = e^-4 × 4^2 / 2! = 0.1465, or 14.65%. At-most mode sums probabilities from zero through k. At-least mode uses the complement of the cumulative value through k - 1.

Understanding Poisson Probability For Interval Counts

The Poisson distribution gives the probability of a whole-number count in a fixed interval. The span may be one hour, one mile, one batch, or some other set range. Enter the mean count lambda and the target count k. Exact mode evaluates the probability of exactly k events. At-most mode adds probabilities from zero through k. At-least mode takes one minus the sum through k - 1. These modes match words such as exactly, no more than, and at least. The model assumes a stable mean rate and independent event occurrence. Calls, defects, and arrivals may fit those assumptions. The answer is a value from zero to one and can also be shown as a percent. Each mode uses the same probability mass function.

Poisson Probability: The Formula Behind The Result

The Poisson probability mass function calculates the likelihood of a given number of events occurring in a fixed interval of time or space.

  • P(X = k) = e^(−λ) × λᵏ ÷ k!.
  • At most: P(X ≤ k) = Σ P(X = i) for i = 0 to k.
  • At least: P(X ≥ k) = 1 − P(X ≤ k − 1).

λ: expected mean count in the interval (lambda).

k: target nonnegative whole-number count.

e: the natural exponential constant.

k!: factorial, equal to k × (k − 1) × ... × 1.

Poisson Calculator Steps: Enter, Calculate, Then Check

Inputs

  • Mean rate lambda: enter the expected count for the same interval.
  • Number of events: enter a nonnegative whole number.

Steps

  1. Match the mean rate to the problem's interval.
  2. Enter lambda.
  3. Enter the event count k.
  4. Select the wording that matches the question.
  5. Calculate and check the zero-to-one range.

Poisson Probability Example With Every Step Shown Clearly

Mean arrivals rate lambda is 4 per hour, target event count k is 2 arrivals in the next hour, in exact mode.

  1. Write the formula: P(X = 2) = e^−4 × 4² ÷ 2!.
  2. Find the power term: 4² = 16.
  3. Find the factorial: 2! = 2 × 1 = 2.
  4. Substitute back: P(X = 2) = e^−4 × 16 ÷ 2 = e^−4 × 8.
  5. Evaluate: e^−4 ≈ 0.0183156, giving 0.0183156 × 8 ≈ 0.146525.
  6. Convert to a percentage: 0.146525 × 100 = 14.6525%.

P(X = 2) = 14.65%, rounded to two decimal places. (For lambda = 4, there is a 14.65% chance of exactly two arrivals).

When Poisson Event Probabilities Give A Good Model

Use this tool for counts in a fixed span when the mean rate is known and events act on their own. Match the mode to the words exact, at most, or at least.

For other statistical, cumulative distribution, and continuous rate calculators, visit the math calculators hub and look up concepts in the glossary.

Assumptions

  • Event counts are nonnegative whole numbers.
  • Lambda is positive and matches the stated interval.
  • The average rate stays stable within that interval.
  • Events occur independently under the model.

Limitations

  • Strong clustering or a changing rate can make the model unsuitable.
  • The calculator does not test whether Poisson assumptions fit observed data.
  • Lambda must use the same interval as the count question.
  • Very large inputs may be affected by numerical precision.

In Practice

The main error is mixing at least with at most. Write the inequality first. For P(X >= k), subtract the sum through k - 1, not through k.

Related Calculators

Normal CDF Calculator

Find the chance below, above, or between values on a normal curve from its mean, spread, and one or two cutoffs for each result.

Open the Calculator

Exponential Growth Calculator

Find how a quantity grows or shrinks with a discrete or continuous rate, using a start value, rate, and number of time periods.

Open the Calculator

Binomial Expansion Calculator

Expand (a + b)^n with the binomial theorem, list each coefficient and power, and compare the result with Pascal's triangle.

Open the Calculator

Related Guides

Poisson Distribution Calculator Questions And Answers

What Is The Difference Between Exact And At Most?

Exact and at-most modes give different answers for the same event count. At most gives P(X <= k), so the tool adds zero, one, and each count through k. The at-most value cannot be less than the exact value for the same k.

How Do You Calculate An At-Least Probability?

For at least k, subtract the chance of counts below k from one. This is the short way to handle the long right tail. Subtracting through k would give the chance of more than k, not at least k.

When Is A Poisson Distribution Appropriate?

Use a Poisson model for counts in a fixed span when the mean rate is steady and events act on their own. OpenStax gives count cases over time or space. A rate that shifts a lot may need a different model.

Is Poisson The Same As The Exponential Distribution?

Poisson counts events, while the exponential model measures the wait between events. An exponential model measures the wait from one event to the next in a linked process. One uses whole counts, while the other uses continuous time.

Sources

Last updated: . Reviewed for accuracy against the formula shown above.