Question

A bank offers loan to its customers on different types of interest namely, fixed rate, floating rate and variable rate. From the past data with the bank, it is known that a customer avails loan on fixed rate, floating rate or variable rate with probabilities 10%, 20% and 70% respectively. A customer after availing loan can pay the loan or default on loan repayment. The bank data suggests that the probability that a person defaults on loan after availing it at fixed rate, floating rate and variable rate is 5%, 3% and 1% respectively.

Based on the above information, answer the following:

1. What is the probability that a customer after availing the loan will default on the loan repayment?   [2]
2. A customer after availing the loan, defaults on loan repayment. What is the probability that he availed the loan at a variable rate of interest?   [2]

Answer 1

Let E₁: Loan at fixed rate

E₂: Loan at floating rate

E₃: Loan at variable rate

`P(E_1) = 10/100`

= `1/10`

`P(E_2) = 20/100`

= `2/10`

`P(E_3) = 70/100`

= `7/10`

Event A be a person defaults loan after availing.

`P(A//E_1) = 5/100`

`P(A//E_2) = 3/100`

`P(A//E_3) = 1/100`

(i) P(A) = P(E₁) P(A/E₁) + P(E₂) P(A/E₂) + P(E₃) P(A/E₃)

= `1/10 xx 5/100 + 2/10 xx 3/100 + 7/10 xx 1/100`

= `5/1000 + 6/1000 + 7/1000`

= `(5 + 6 + 7)/1000`

= `18/1000`

= 0.018

(ii) `P(E_3//A) = (P(E_3) P(A//E_3))/(P(E_1) P(A//E_1) + P(E_2)P(A//E_2) + P(E_3) P(A//E_3)`

= `(7/10 xx 1/100)/(18/1000)`

= `(7/1000)/(18/1000)`

= `7/18`