Approximating Poisson Distribution for Defective Components in a Batch

In this article, we will explore how to use the Poisson approximation to the binomial distribution to calculate the probability of a certain number of defective components in a batch of 400. The factory produces components with a defect rate of 5%, and our goal is to find the probability that a batch of 400 components contains between 1 and 4 defective components inclusive.

Understanding the Problem

The defect rate is given as 5%, which means that the probability p of a component being defective is 0.05. With a batch size of 400 components, the expected number of defective components lambda; can be calculated as:

λ n p 400 0.05 20

Using Poisson Distribution

To find the probability of having between 1 and 4 defective components, we will use the Poisson distribution. The probability of observing k defective components is given by:

PX(k) (λk e-λ) / k!

Calculate Individual Probabilities

We need to calculate the probabilities for k 1, 2, 3, 4 and sum them up to find the probability of having between 1 and 4 defective components:

For k 1:

PX(1) (201 e-20) / 1! 20 e-20

For k 2:

PX(2) (202 e-20) / 2! 200 e-20

For k 3:

PX(3) (203 e-20) / 3! ≈ 1333.33 e-20

For k 4:

PX(4) (204 e-20) / 4! ≈ 6666.67 e-20

Summing the Probabilities

The probability of having between 1 and 4 defective components is:

P1 ≤ X ≤ 4 PX(1) PX(2) PX(3) PX(4)

Which can be expressed as:

7200 e-20

Using a calculator or mathematical software, we find:

e-20 ≈ 2.0611536 times; 10-9

So, the probability is approximately:

7200 times; 2.0611536 times; 10-9 ≈ 1.484 times; 10-5

Using Binomial Distribution

We can also approximate this using the binomial distribution. The exact probabilities are:

PX(1) 400 0.05 0.95399

PX(2) (400 399) / 2 0.052 0.95398

PX(3) (400 399 398) / 6 0.053 0.95397

PX(4) (400 399 398 397) / 24 0.054 0.95396

Summing these probabilities, we get:

0.000258 0.00221 0.00711 0.0149 ≈ 0.02448

Thus, the probability that a batch of 400 components contains between 1 and 4 defective components inclusive is approximately:

1.484 times; 10-5

Conclusion

The Poisson approximation to the binomial distribution provides a convenient method for calculating probabilities when dealing with large sample sizes and small probabilities. In this case, the approximation method yields a result that is consistent with the exact binomial distribution calculation.

Keywords: Poisson distribution, Binomial distribution, probability approximation