The Poisson approximation to the binomial

The approximation

If X ~ B(n, p) with n large and p small, then X can be approximated by Po(np).

The specification gives no fixed cut-off; a rough guide is n > 50 and p < 0.2, but the mark scheme wording is simply 'n is large and p is small'. When justifying the approximation you need both conditions; examiners report that giving only one loses the mark. 'The mean is close to the variance' is not accepted as a justification.

Why it works

B(n, p) has mean np and variance np(1 - p). When p is small, 1 - p is close to 1, so the variance is close to np: the mean and variance are nearly equal, as for a Poisson distribution with λ = np.

If p is close to 1

Count failures. If 97% of 150 components work, the number that fail is B(150, 0.03), which is approximately Po(4.5).

If a question says 'use a suitable approximation', an exact binomial answer usually loses marks, so use the Poisson distribution as asked.

Worked example

2% of items from a machine are faulty. In a sample of 200, use a suitable approximation to find the probability that at most 3 are faulty.

1. X ~ B(200, 0.02); n is large and p is small, so X ≈ Po(4).
2. P(X ≤ 3) from the Po(4) table

Answer: 0.4335 (the exact binomial value is 0.4315)

Check yourself

1. X ~ B(200, 0.02). Use a Poisson approximation to find P(X ≤ 3). Give your answer to 4 decimal places.
   Show answer

   0.4335. λ = 200 × 0.02 = 4. P(X ≤ 3) from the Po(4) table = 0.4335.

2. X ~ B(400, 0.005). State the value of λ for a Poisson approximation.
   Show answer

   2. λ = np = 400 × 0.005 = 2.

3. X ~ B(300, 0.01). Use a Poisson approximation to find P(X = 5). Give your answer to 4 decimal places.
   Show answer

   0.1008. Po(3): e^-3 × 3⁵ / 5! = 0.1008.

4. 97% of components work. In a box of 150, use a suitable approximation to find the probability that at least 148 work. Give your answer to 4 decimal places.
   Show answer

   0.1736. Failures F ~ B(150, 0.03) ≈ Po(4.5). At least 148 work means F ≤ 2: P(F ≤ 2) = 0.1736.

5. X ~ B(120, 0.025). Use a Poisson approximation to find P(X > 4). Give your answer to 4 decimal places.
   Show answer

   0.1847. Po(3): 1 - P(X ≤ 4) = 1 - 0.8153 = 0.1847.

6. Which binomial distribution is best approximated by a Poisson distribution?
   • B(20, 0.4)
   • B(10, 0.05)
   • B(100, 0.5)
   • B(500, 0.004)
   Show answer

   B(500, 0.004). n is large and p is small.

7. Which justification of a Poisson approximation to B(200, 0.02) earns the mark?
   • The mean is close to the variance
   • n is large and p is small
   • n is large
   • p is less than 0.5
   Show answer

   n is large and p is small. Mark schemes require both conditions, n large and p small.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S2
• Pearson Edexcel IAL Statistics S2 (WST02) examiners' report, summer 2025
• Pearson Edexcel IAL Statistics S2 (WST02) mark scheme, June 2025 (copy hosted by Physics and Maths Tutor)

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