Question

2. A loan officer is considering a loan request from a customer. Based on the bank’s records over years, there is an 8% chance that a customer who has overdrawn an account will default on the loan.

However, there is only a 1% chance that a customer who has never overdrawn an account will default on the loan. Based on the customer’s credit history, the loan officer believes there is a 40% chance that this customer will overdraw his account. Let O be the event that customer overdraws his account, and let D be the event that the customer defaults on the loan. Use tree diagram to answer b) and c).

1. Are the event of default on the loan independent of the event of overdraw?

2. What is the probability that the customer will default on the loan

3. Given that a customer has defaulted on his loan, what is the probability that he overdraws the account?

Answer 1

(a)

The value of P\left( {D \cap O} \right)P(D∩O) is obtained below:

Define the events:

Let D be the event that the customer defaults on the loan

Let O be the event that customer overdraws his account

From the given information,

P(D|O) = 0.08,P(D|Oˉ)=0.01,P(O)=0.4.

P(D|O) = P(D∩O)/ P(O)

P(D|O) = P(D|O)P(O)

= 0.08×0.40

= 0.032​

The value of the value of P(D) is obtained below:

P(Oˉ) = 1−P(O)

= 1−0.4

= 0.6

P(D∣Oˉ) = P(D∩Oˉ) / P(Oˉ)

P(D∩Oˉ) = P(D∣Oˉ)×P(Oˉ)

= 0.01×0.6

= 0.006​

P(D∩Oˉ) = P(D)−P(D∩O)

P(D) = P(D∩Oˉ)+P(D∩O)

= 0.006+0.032

= 0.038​

The independence of default on the loan and overdraw is obtained below:

Consider,

P(D)×P(O) = 0.038×0.40

= 0.0152

≠ P(D∩O)​

Hence, the condition P(D∩O)=P(D)×P(O) is not satisfied. Therefore, the events D and O are not independent.

Thus, the event default on the loan is not independent on the overdraw.