A bank offers loans to its customers on different types of interest rates 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 a 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:
(i) What is the probability that a customer after availing the loan will default on the loan repayment?
(ii) 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?
Show Hint
To solve probability problems involving conditional probabilities, use the law of total probability for finding the total probability and Bayes' theorem for finding conditional probabilities.
The following probabilities are provided:
- Probability of taking a fixed-rate loan: \( P(F) = 0.1 \)
- Probability of taking a floating-rate loan: \( P(Fl) = 0.2 \)
- Probability of taking a variable-rate loan: \( P(V) = 0.7 \)
- Probability of defaulting given a fixed-rate loan: \( P(D|F) = 0.05 \)
- Probability of defaulting given a floating-rate loan: \( P(D|Fl) = 0.03 \)
- Probability of defaulting given a variable-rate loan: \( P(D|V) = 0.01 \)
(i) What is the probability a customer will default on their loan?
Using the law of total probability, the overall probability of default \( P(D) \) is calculated as:
\[P(D) = P(D|F) \cdot P(F) + P(D|Fl) \cdot P(Fl) + P(D|V) \cdot P(V).\]
Substituting the given values:
\[P(D) = (0.05 \times 0.1) + (0.03 \times 0.2) + (0.01 \times 0.7)\]
\[P(D) = 0.005 + 0.006 + 0.007 = 0.01.\]
Therefore, the probability of a customer defaulting on their loan is \( 0.01 \) or 1%.
(ii) Given a customer defaulted on their loan, what is the probability they took a variable-rate loan?
We need to calculate \( P(V|D) \), the probability of having taken a variable-rate loan given a default. Bayes' theorem is applied here:
\[P(V|D) = \frac{P(D|V) \cdot P(V)}{P(D)}.\]
Substituting the values:
\[P(V|D) = \frac{0.01 \times 0.7}{0.01} = \frac{0.007}{0.01} \approx 0.39.\]
Thus, the probability that a customer took a variable-rate loan, given that they defaulted, is approximately \( 0.39 \) or 3.9%.
