SOLUTION: Recent difficult economic times have caused an increase in the foreclosure rate of home mortgages. Statistics from the Penn Bank and Trust Company show their monthly foreclosure ra

Question 1192090: Recent difficult economic times have caused an increase in the foreclosure rate of home mortgages. Statistics from the Penn Bank and Trust Company show their monthly foreclosure rate is now one loan out of every 136 loans. Last month the bank approved 300 loans.
a. How many foreclosures would you expect the bank to have last month? (Round your answer to 4 decimal places.)
b. What is the probability of exactly two foreclosures? (Round your answer to 4 decimal places.)
c. What is the probability of at least one foreclosure? (Round your answer to 4 decimal places.)

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Here's how to solve this problem:
**a. Expected Number of Foreclosures:**
* The foreclosure rate is 1/136.
* The bank approved 300 loans.
* Expected foreclosures = (Foreclosure rate) * (Number of loans)
* Expected foreclosures = (1/136) * 300 ≈ 2.2059
So, you would expect approximately 2.2059 foreclosures.
**b. Probability of Exactly Two Foreclosures:**
This is a binomial probability problem. We can approximate it with the Poisson distribution since n is large and p is small.
* n = 300 (number of trials/loans)
* p = 1/136 (probability of foreclosure)
* x = 2 (number of foreclosures we're interested in)
* λ = n * p = 300 * (1/136) ≈ 2.2059 (average number of foreclosures)
The Poisson probability formula is:
P(x) = (e^(-λ) * λ^x) / x!
P(2) = (e^(-2.2059) * 2.2059^2) / 2!
P(2) ≈ (0.1104 * 4.866) / 2
P(2) ≈ 0.2687
The probability of exactly two foreclosures is approximately 0.2687.
**c. Probability of At Least One Foreclosure:**
The easiest way to calculate this is to use the complement rule. The probability of at least one foreclosure is equal to 1 minus the probability of *no* foreclosures.
P(at least one) = 1 - P(0)
Using the Poisson formula for P(0):
P(0) = (e^(-2.2059) * 2.2059^0) / 0!
P(0) ≈ 0.1104
Therefore:
P(at least one) = 1 - 0.1104
P(at least one) ≈ 0.8896
The probability of at least one foreclosure is approximately 0.8896.

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