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Poisson probabilities, tables and changing the interval

Edexcel IAL Mathematics · WST02 Statistics 2

Cumulative Poisson tables work just like the binomial ones. The key Poisson skill is the additive property: if you change the length of the interval, you scale λ in proportion.

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Cumulative probabilities

The Poisson table gives P(X ≤ x) to 4 d.p. for chosen values of λ; a calculator gives the same. Use the same rewriting rules as for the binomial: P(X ≥ x) = 1 - P(X ≤ x - 1), P(a < X ≤ b) = P(X ≤ b) - P(X ≤ a).

The additive property

If X, the number of events in one minute, is Po(λ), then the number in t minutes is Po(tλ). Always rewrite λ for the interval in the question.

Example: 4 calls per hour on average. In 15 minutes, λ = 4 × 15/60 = 1; in 2 hours, λ = 8.

The same idea means independent Poisson counts can be added: events in two separate intervals with means 3 and 5 give a total that is Po(8).

Intervalλ
1 hour4
15 minutes1
30 minutes2
2 hours8

Several intervals

Questions combine Poisson with other ideas. For 'at least one call in each of three separate 15-minute periods', find p = P(X ≥ 1) = 1 - e-1 for one period, then cube it, because the periods are independent. For 'exactly 2 of 5 periods', use B(5, p).

Worked example

A helpline receives calls at a rate of 4 per hour. Find the probability of more than 10 calls in 2 hours.

  1. In 2 hours, X ~ Po(8).
  2. P(X > 10) = 1 - P(X ≤ 10)
  3. = 1 - 0.8159

Answer: 0.1841

Key idea: Tables give P(X ≤ x). Scale λ with the length of the interval: Po(λ) per unit means Po(tλ) per t units.

Check yourself

  1. X ~ Po(6). Find P(X ≤ 4). Give your answer to 4 decimal places.
    Show answer

    0.2851. From the table: 0.2851.

  2. X ~ Po(6). Find P(X ≥ 8). Give your answer to 4 decimal places.
    Show answer

    0.2560. 1 - P(X ≤ 7) = 1 - 0.7440 = 0.2560.

  3. X ~ Po(6). Find P(3 < X ≤ 7). Give your answer to 4 decimal places.
    Show answer

    0.5928. P(X ≤ 7) - P(X ≤ 3) = 0.7440 - 0.1512 = 0.5928.

  4. Calls arrive at a rate of 4 per hour. Find the probability of no calls in a 15-minute period. Give your answer to 4 decimal places.
    Show answer

    0.3679. In 15 minutes λ = 1, so P(X = 0) = e-1 = 0.3679.

  5. Calls arrive at a rate of 4 per hour. Find the probability of more than 10 calls in 2 hours. Give your answer to 4 decimal places.
    Show answer

    0.1841. Po(8): 1 - P(X ≤ 10) = 1 - 0.8159 = 0.1841.

  6. Calls arrive at a rate of 4 per hour. Find the probability that there is at least one call in each of three separate 15-minute periods. Give your answer to 4 decimal places.
    Show answer

    0.2526. For one period P(X ≥ 1) = 1 - e-1 = 0.63212; for three independent periods, cube it: 0.2526.

  7. X ~ Po(6). Find the smallest value of k such that P(X ≤ k) ≥ 0.99.
    Show answer

    12. From the table, P(X ≤ 11) = 0.9799 and P(X ≤ 12) = 0.9912, so k = 12.

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Last updated 8 November 2026. Spotted a mistake? Tell us.