A manufacturer ships toasters in cartons of 50. In each carton, they estimate a 2% chance that one of the toasters will need to be sent back for minor repairs. In a batch of 25,000 toasters, what are the chances that fewer than 475 need to be returned?

Question

The probability that fewer than 475 toasters need to be returned is...

Answer 1

To solve this problem, we can use the binomial probability formula:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

where n is the number of trials (toasters), k is the number of successes (toasters needing repairs), and p is the probability of success (2% chance of needing repairs).

We want to calculate the probability that X < 475, which means we need to find the sum of the probabilities for k = 0 to k = 474.

First, let's find the expected value (mean) and standard deviation of this binomial distribution:

Expected value (mean) = n * p = 25000 * 0.02 = 500
Standard deviation = sqrt(n * p * (1-p)) = sqrt(25000 * 0.02 * 0.98) ≈ 22.07

Now, we'll use the normal distribution to approximate the binomial distribution. We'll convert X = 474.5 (we add 0.5 for continuity correction) to a z-score:

z = (X - mean) / standard deviation = (474.5 - 500) / 22.07 ≈ -1.156

Now, we can use a standard normal table or calculator to find the probability that z < -1.156, which gives us the probability that fewer than 475 toasters need to be returned.

Using a standard normal table or calculator, we find that P(z < -1.156) ≈ 0.1241.

Therefore, the probability that fewer than 475 toasters need to be returned is approximately 12.41%.