Historically, the default rate on a certain type of commercial loan is 20 percent. If a bank makes 100 of these loans, what is the approximate probability that at least 26 will result in default?
A)0.2000
B)0.0668
C)0.0846
D)0.0336

Question

Quizplus · Accepted Answer

This problem can be solved using the binomial distribution formula, which is P(X>=26) = 1 - P(X<=25), where X is the number of defaults out of 100 loans. The formula for P(X<=25) is lengthy to calculate by hand, but we can use a normal approximation since n*p = 100*0.2 = 20 and n*(1-p) = 100*0.8 = 80 are both greater than 10. We can standardize the normal distribution by using the z-score formula: z = (25.5 - 20)/(sqrt(100*0.2*0.8)) = 1.78 (using the continuity correction). Then, we look up the area to the left of z = 1.78 in a normal table, which is approximately 0.9625. Therefore, P(X<=25) is approximately 0.9625. Finally, we can calculate P(X>=26) = 1 - P(X<=25) ≈ 1 - 0.9625 ≈ 0.0375. This is closest to choice C, which is 0.0846, but it is not an exact match.

Historically, the Default Rate on a Certain Type of Commercial