Math 362 - Computer Lab #6 - Fall 2002

Introduction to Probability Theory

Computer Lab 6: Conditional Probabilities and Bayes's Theorem


This lab is based on the first project of the course Mathematics for Business Decisions I by Richard B. Thompson and Christopher G. Lamoureux.

The file Loan_Records.MTW contains information about 8226 bank customers who took out a commercial loan with one of the following banks: the "DuPont Bank", the "Cajun Bank" or the "BR Bank". The first column in the file is the borrower number, the second contains the name of the bank, the third column is the number of years in business the borrower had when his/her business had difficulties and he/she had to work out an arrangement with the bank, the fourth column is the borrower's education level, the fifth column describes the state of the economy at the time of the loan work out, and the last column says whether or not the loan was paid back in full by the borrower. The data file is incomplete in the sense that the different banks kept different records. More precisely, the DuPont bank has information about the state of the economy, the Cajun Bank has information on its customers' level of education, and the BR bank has information on its customers' number of years in business.
In the first part of this lab, we will use the data file to estimate various probabilities. In the second part, we will use these probabilities to decide whether a bank should work out an arrangement with or foreclose on a borrower whose business is currently illiquid.

Before proceeding we define the following events:


Part One

  1. Download the file and open it with MINITAB, using the File -> Open Worksheet... command.

  2. Use Stat -> Tables -> Tally... to estimate the following probabilities:

    • Pr(S)

    • Pr(F)

    • Pr(Y)

    • Pr(T)

    • Pr(C)

    Note that for the last three cases, we assume that each probability can be estimated using a restricted data set obtained by limiting ourselves to the records of a particular bank.

  3. Use Stat -> Tables -> Cross Tabulation... to find the following conditional probabilities. Note that you will have to add an asterisk (*) in the empty cells of the last row of data, to prevent MINITAB from thinking that some of the columns are shorter than the others.

    • Pr(Y | S) (this is the probability that a borrower has 7 years of experience, given that he/she paid back the loan in full)

    • Pr(T | S)

    • Pr(C | S)

    • Pr(Y | F)

    • Pr(T | F)

    • Pr(C | F)

    • Pr(S | Y)

    • Pr(F | Y)

  4. Two events A and B are independent if and only if Pr(AB) = Pr(A) Pr(B). A similar definition applies when the events are conditioned to another event. Assuming that the events Y, T, and C are independent, even when they are conditioned to S and F, estimate the following probabilities:

    • Pr(Y T C | S)


    • Pr(Y T C | F)


  5. Draw a picture of the sample space S which contains the events S, F, and Y T C.

















    Use the picture to obtain a formula for Pr(Y T C) in terms of Pr(Y T C | S), Pr(Y T C | F), P(S) and P(F).







  6. Use Bayes's theorem to calculate the following probabilities:

    • Pr(S | Y T C)






    • Pr(F | Y T C)







Part Two

Assume that you are a loan officer and that the bank you work for has an outstanding commercial loan with a full value of $4,000,000. If the bank decides to foreclose on the borrower, it will recover $2,100,000. If the bank decide to work out an arrangement, either the work out will be successful, in which case the bank will get the $4,000,000 back, or the work out will fail, in which case the bank will only recover $250,000. You also know that the borrower has a Bachelor's degree, that he has 7 years of business experience, and that the economy is in normal times. Using this information and the probabilities calculated in Part One, decide whether or not the bank should work out an arrangement with this particular borrower.























Reading Assignment: Sections 2.1 through 2.3.


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