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CIE A-Level Maths Study Notes

5.1.4 Poisson Approximation to the Binomial Distribution

The Poisson approximation to the binomial distribution offers a simplified method for calculating probabilities in specific scenarios. It is particularly useful in situations where the binomial distribution parameters meet certain conditions.

What is Poisson Approximation?

  • A method to simplify probability calculations.
  • Used when dealing with a large number of trials and a small chance of success.

When to Use It?

  • Large Number of Trials: Generally, more than 50.
  • Small Probability of Success: The event should be rare.
  • Product of Trials and Probability (np): Should be less than 5. This becomes the mean λλ in Poisson distribution.

Examples

Example 1: Factory Defects (Poisson Distribution)

Consider a factory where the probability of producing a defective component is 0.004, and the daily production is 1000 components. Find the probability of exactly 3 defective components being produced on a given day.

Solution:

  • Poisson Mean (λ): λ = n(p) = 1000(0.004) = 4
  • Probability of 3 Defects: Using Poisson formula, P(X=3)=eλ×λ33!=e4×433!=0.1954P(X = 3) = \frac{e^{-\lambda} \times \lambda^3}{3!} = \frac{e^{-4} \times 4^3}{3!} = ≈ 0.1954
  • Result: Probability 19.54≈ 19.54%
Factory Defects (Poisson Distribution) Graph

Example 2: Comparison with Binomial Probability

Using the same factory scenario, compare the Poisson approximation probability with the exact binomial probability for 3 defective components.

Solution:

  • Binomial Probability: Using binomial formula, $P(X = 3) = \binom{n}{x} \times p^x \times (1 - p)^{n - x} = \binom{1000}{3} \times 0.004^3 \times (1 - 0.004)^{997} = ≈ 0.1956 </li><li><strong>Result:</strong>Probability</li><li><strong>Result:</strong> Probability ≈ 19.56%$
  • Comparison: Poisson (19.54%) is very close to Binomial (19.56%).
Comparison with Binomial Probability Graph
Dr Rahil Sachak-Patwa avatar
Written by: Dr Rahil Sachak-Patwa
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Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.

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