MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2025: question 38

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section E, question 38, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q38 · 4 marks · Case-basedCompetency-basedProbability

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 :

(i)
What is the probability that a customer after availing the loan will default on the loan repayment ?
2 marks
(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 ?
2 marks

Figure in the paper: Illustration of stacks of coins decreasing in height with an arrow labelled 'INTEREST RATES' and a % sign. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Define events for the three loan types and for defaulting, and convert the percentages to probabilities.

Hint 2

For (i), add the products P(type) × P(default | type) over all three types.

Hint 3

For (ii), divide the variable-rate term by the total found in (i) (Bayes' theorem).

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Define the events:
- $E_{1}$ : the loan is at fixed rate
- $E_{2}$ : the loan is at floating rate
- $E_{3}$ : the loan is at variable rate
- $A$ : the customer defaults
Then $P (E_{1}) = \frac{1}{10}, P (E_{2}) = \frac{2}{10}, P (E_{3}) = \frac{7}{10}$ and $P (A ∣ E_{1}) = \frac{5}{100}, P (A ∣ E_{2}) = \frac{3}{100}, P (A ∣ E_{3}) = \frac{1}{100}$ .
(i) By the theorem of total probability:
$P (A) = P (E_{1}) P (A ∣ E_{1}) + P (E_{2}) P (A ∣ E_{2}) + P (E_{3}) P (A ∣ E_{3})$
$= \frac{1}{10} \cdot \frac{5}{100} + \frac{2}{10} \cdot \frac{3}{100} + \frac{7}{10} \cdot \frac{1}{100}$
1 mark
(i) $P (A) = \frac{5 + 6 + 7}{1000} = \frac{18}{1000} = \frac{9}{500}$
1 mark
(ii) By Bayes' theorem:
$P (E_{3} ∣ A) = \frac{P ( E _{3} ) P ( A ∣ E _{3} )}{P ( A )} = \frac{\frac{7}{10} \cdot \frac{1}{100}}{\frac{18}{1000}}$
1 mark
(ii) $P (E_{3} ∣ A) = \frac{7}{18}$
1 mark

Answer: (i) $\frac{9}{500}$ (= 0.018); (ii) $\frac{7}{18}$

Where the marks are

(i) Applies total probability with the correct values (1 mark)
(i) Obtains $\frac{18}{1000} = \frac{9}{500}$ (1 mark)
(ii) Applies Bayes' theorem correctly (1 mark)
(ii) Obtains $\frac{7}{18}$ (1 mark)

Revise the chapter

Chapter: Probability (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Probability, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2025: question 38

Hints

Step-by-step solution

Revise the chapter

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