Question

Records over the past year show that 1 out of 360 loans made by Mammon Bank have defaulted. Find the probability that 3 or more out of 270 loans will default. Hint: Is it appropriate to use the Poisson approximation to the binomial distribution? (Round λ to 1 decimal place. Use 4 decimal places for your answer.)

Answer 1

Answer 2

To determine whether it is appropriate to use the Poisson approximation to the binomial distribution, we need to check if the conditions for using the Poisson distribution are met. The Poisson approximation to the binomial distribution is valid when the number of trials (n) is large, and the probability of success (p) is small.

In this case, we have n = 270 (number of loans) and p = 1/360 (probability of defaulting on a loan). Let's calculate λ (the average number of defaults) using the Poisson approximation formula:

λ = n * p

λ = 270 * (1/360) ≈ 0.75

Now, to find the probability that 3 or more out of 270 loans will default, we can use the Poisson distribution formula:

P(X ≥ 3) = 1 - P(X < 3)

Using the Poisson probability formula, we can calculate the probabilities for X = 0, 1, and 2 defaults, and then subtract that sum from 1 to find the probability of 3 or more defaults:

P(X = 0) = (e^(-λ) * λ^0) / 0! ≈ 0.4724 P(X = 1) = (e^(-λ) * λ^1) / 1! ≈ 0.3543 P(X = 2) = (e^(-λ) * λ^2) / 2! ≈ 0.1329

Now, calculate the probability of 3 or more defaults:

P(X ≥ 3) = 1 - (P(X = 0) + P(X = 1) + P(X = 2)) P(X ≥ 3) ≈ 1 - (0.4724 + 0.3543 + 0.1329) P(X ≥ 3) ≈ 1 - 0.9596 P(X ≥ 3) ≈ 0.0404

Therefore, the probability that 3 or more out of 270 loans will default is approximately 0.0404, or 4.04% (rounded to 4 decimal places).