Poisson distribution calculator

A Poisson distribution calculator gives the probability of a number of random events happening in a fixed time or space, when you know the average rate λ (lambda). In our Probability Calculator, choose Poisson, type λ, pick the region you want, such as exactly 2 or at most 2, and it shades the bars and gives the probability.

Poisson is one of the distributions in the Probability Calculator, so the first step is to switch to it. Everything below uses that mode.

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How to use the Poisson distribution calculator

1. Open the calculator and, under Distribution, choose Poisson. It starts on Normal.
2. Type the average rate in Mean rate λ. It must be positive, and it can be a decimal such as 2.5.
3. In Shade, choose the region: P(X < a) for at most a, P(X > a) for at least a, P(a < X < b) for a range, or P(X < a or X > b) for both tails.
4. Type Bound a, and Bound b for a range, or drag the boundary lines on the graph.
5. Read the answer in Probability, written as a statement such as P(X ≤ 2) = 0.2381.

Because a Poisson count is a whole number, the calculator includes the bound: the P(X < a) option gives P(X ≤ a) and the P(X > a) option gives P(X ≥ a). The answer line always shows which. The Formulas section shows the Poisson formula, the mean and variance for your λ, and P(X ≥ 1). Practice and help includes Poisson questions with worked solutions.

Turning the question into a region

The Poisson distribution formula

If X counts events that happen at random at an average rate of λ in a given interval, X ~ Po(λ), and the probability of exactly k events is:

The mean and the variance are both λ, so the standard deviation is √λ. Probabilities such as P(X ≤ k) are found by adding the single values from 0 up to k, which is what the calculator does for you.

When to use a Poisson model

• Events happen one at a time, not in clumps.
• Events happen independently of each other.
• They happen at a constant average rate over the interval.
• You count how many happen in a fixed interval of time, distance, area or volume, and there is no fixed maximum.

Typical examples are emails arriving in an hour, typing errors per page, or radioactive decays per second. If there is a fixed number of trials, use the binomial distribution instead.

Worked examples you can check in the calculator

An inbox receives emails at random, at an average of 4 an hour. So X ~ Po(4) for the number of emails in an hour.

Example 1: exactly k

What is the probability of exactly 2 emails in an hour?

Set λ = 4, shade P(a < X < b) with a = 2 and b = 2. The calculator gives P(2 ≤ X ≤ 2) = 0.1465.

Example 2: at most k

What is the probability of at most 2 emails in an hour? Add the probabilities of 0, 1 and 2:

Shade P(X < a) with a = 2: the calculator gives P(X ≤ 2) = 0.2381.

Example 3: at least k

What is the probability of at least one email? Use the complement, 1 minus the chance of none:

Shade P(X > a) with a = 1, or read it straight from the P(X ≥ 1) card in the Formulas section. For more than 6 emails, shade P(X > a) with a = 7: P(X ≥ 7) = 0.1107.

Example 4: changing the interval

What is the probability of no emails in a 30-minute meeting? The rate is 4 an hour, so for half an hour λ = 2:

Scale λ to match the interval in the question before you start: 4 an hour is 2 per half hour and 96 per day. Set λ = 2 with a = 0 and b = 0 to check.

Check you have the idea:

The Poisson approximation to the binomial

When a binomial distribution has many trials and a small chance of success, a Poisson distribution with λ = np gives almost the same answers. For example, if 1% of items are faulty and you check 200, the number faulty is B(200, 0.01), and λ = 200 × 0.01 = 2.

The two agree to 2 decimal places. The approximation works best when n is large and p is small, so that np is moderate.

Common mistakes

• Using the wrong rate. Scale λ to the interval asked about. A rate per hour is not the rate per minute.
• Mixing up "fewer than" and "at most". Fewer than 2 is X ≤ 1. More than 6 is X ≥ 7.
• Forgetting P(X = 0). "At most 2" includes 0, so add the terms for 0, 1 and 2.
• Typing a decimal bound. The bounds are whole numbers. The calculator rounds a decimal bound down, so 1.5 is read as 1.
• Using Poisson with a fixed number of trials. A count with a maximum, such as heads in 10 coin tosses, is binomial.
• Taking the standard deviation as λ. The variance is λ. The standard deviation is √λ.

Practise probability distributions in a short timed paper

Our daily papers do not include Poisson questions yet, but AP Statistics and A-level Maths: Statistics papers practise the binomial and normal distributions, which you work out in the same calculator. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks. Our probability calculator guide covers every distribution in the tool.

Read next

Maths · StatisticsProbability calculator: normal, binomial and PoissonFree probability calculator: shade P(X < a), P(X > a) or a range on a normal, binomial or Poisson distribution and read the probability, with examples.