Use Poisson distribution to find the probability of at most 5 detective fuses in a box of 200 fuses. Experience shows that 2% of such fuses are defective.

I. Introduction

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. In this problem, we are using the Poisson distribution to find the probability of at most 5 defective fuses in a box of 200 fuses, given that 2% of such fuses are defective.

II. Problem Analysis

The problem is to find the probability of at most 5 defective fuses in a box of 200 fuses, given that 2% of such fuses are defective. The parameters for the Poisson distribution are λ (the average rate of success), x (the number of successes), and e (a constant approximately equal to 2.71828).

III. Formulas and Calculations

The Poisson distribution formula is:

P(x; λ) = e^(-λ) * (λ^x / x!)

where:

• P(x; λ) is the probability of x successes given an average rate of success λ,
• e is a constant approximately equal to 2.71828,
• λ is the average rate of success, and
• x is the number of successes.

First, we calculate λ, the average rate of success. Given that 2% of fuses are defective, and there are 200 fuses in a box, λ = n * p = 200 * 0.02 = 4.

Next, we calculate the probability for x = 0 to 5 using the formula and sum up the results to get the probability of at most 5 defective fuses.

IV. Programming Solution (optional)

If required, we can use the scipy.stats.poisson module in Python to calculate the probability. Here is a sample program:

from scipy.stats import poisson

# average rate of success
lambda_ = 4

# calculate the probability of at most 5 defective fuses
prob = sum(poisson.pmf(k, lambda_) for k in range(6))

print(f"The probability of at most 5 defective fuses is {prob:.4f}")

V. Conclusion

The Poisson distribution is a powerful tool in probability theory that allows us to calculate the probability of a given number of events occurring in a fixed interval of time or space. In this problem, we used the Poisson distribution to find the probability of at most 5 defective fuses in a box of 200 fuses, given that 2% of such fuses are defective. The limitations of the Poisson distribution include the assumption of a constant average rate of success and the independence of events.

VI. Examples (optional)

If needed, we can provide additional examples of similar problems solved using the Poisson distribution. For example, we could calculate the probability of at most 3 defective fuses in a box of 100 fuses, given that 1% of such fuses are defective.

Summary

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. In this problem, we are using the Poisson distribution to find the probability of at most 5 defective fuses in a box of 200 fuses, given that 2% of such fuses are defective.

Analogy

Imagine you have a box of 200 fuses, and you want to find the probability of having at most 5 defective fuses. You know that 2% of the fuses in the box are defective. This problem can be solved using the Poisson distribution, which helps us calculate the probability of a given number of events occurring in a fixed interval of time or space.