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 $λ$ 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, $P(X = 3) = \frac{e^{-\lambda} \times \lambda^3}{3!} = \frac{e^{-4} \times 4^3}{3!} = ≈ 0.1954$
• Result: Probability $≈ 19.54$

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 $</li><li><strong>Result:</strong> Probability$≈ 19.56%$
• Comparison: Poisson (19.54%) is very close to Binomial (19.56%).