During periods of economic recession, experience shows that 10% of the people wi

Question

During periods of economic recession, experience shows that 10% of the people with loans outstanding at a certain commercial bank will become delinquent in their loan repayment schedule. To examine this issue and the characteristics of borrowers from her bank, the bank president randomly selects n = 100 account files at a time in which the economic climate of the community is very uncertain. Find the following probabilities. a. Ten or more of the borrowers are delinquent in their loan repayment schedule. b. No more than 5 arc delinquent.

Explanation / Answer

Solution

Let X = number of borrowers, out of the randomly selected 100, who will be delinquent in loan repayment schedule. Then, X is distributed as Binomial with parameters n ( = 100) and p (= probability of a borrower turning delinquent during economic recession, which is given to be 0.1, i.e., 10%). Probability Mass Function (pdf) of this distribution is given by

p(x) = P(X = x) = (nCx )px(1- p)n - x . These values can be calculated by hand, or using Excel Function or can read off from Binomial Probability Tables.

Calculations given below are based on Excel Function.

Part (a)

P(10 or more delinquent borrowers) = sum (over x = 10 to 100) of {(100Cx )(0.1)x(0.9)100 - x)}.

This can be calculated faster by using complementary probability as

P(10 or more delinquent borrowers) = 1 - P(less than 10 delinquent borrowers)

= 1 - sum (over x = 0 to 9) of {(100Cx )(0.1)x(0.9)100 - x)}.

These values as obtained from Excel Function are given below:

2.66E-05

0.000295

0.001623

0.005892

0.015875

0.033866

0.059579

0.088895

0.114823

0.130416

Total = 0.45129

So,  P(10 or more delinquent borrowers) = 1 - 0.45129 = 0.54871 ANSWER

Part (b)

P(No more than 5 delinquent borrowers) = sum (over x = 0 to 5) of {(100Cx )(0.1)x(0.9)100 - x)}.

These values as obtained from Excel Function are given below:

2.66E-05

0.000295

0.001623

0.005892

0.015875

0.033866

Total = 0.057577 ANSWER

2.66E-05

0.000295

0.001623

0.005892

0.015875

0.033866

0.059579

0.088895

0.114823

0.130416

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