CBSE Class 12 Mathematics Probability Q #700

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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.
COMPETENCY BASED
APPLY
2 Marks 2025 AISSCE(Board Exam) VSA
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?
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Step-by-Step Solution

Let F, L, and V denote the events that a customer avails loan at fixed rate, floating rate, and variable rate, respectively. Let D denote the event that a customer defaults on loan repayment.

We are given:

  • P(F) = 0.10
  • P(L) = 0.20
  • P(V) = 0.70
  • P(D|F) = 0.05
  • P(D|L) = 0.03
  • P(D|V) = 0.01

We want to find P(V|D), the probability that the customer availed the loan at a variable rate given that they defaulted.

Using Bayes' Theorem:

P(V|D) = [P(D|V) * P(V)] / P(D)

First, we need to find P(D), the probability of default. We can use the law of total probability:

P(D) = P(D|F) * P(F) + P(D|L) * P(L) + P(D|V) * P(V)

P(D) = (0.05 * 0.10) + (0.03 * 0.20) + (0.01 * 0.70)

P(D) = 0.005 + 0.006 + 0.007

P(D) = 0.018

Now we can find P(V|D):

P(V|D) = (0.01 * 0.70) / 0.018

P(V|D) = 0.007 / 0.018

P(V|D) = 7/18

Correct Answer: 7/18

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student needs to apply Bayes' Theorem to calculate the conditional probability based on the given scenario.
Knowledge Dimension: CONCEPTUAL
Justification: The question requires understanding of conditional probability and Bayes' theorem to solve the problem. It's not just recalling a formula but applying the concept.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question is designed to assess the student's ability to apply probability concepts to a real-world scenario, rather than simply recalling definitions or formulas.

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