A departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month. Based on the above information, answer the following questions : (i) Let E_1 and E_2 respectively denote the event of customer paying or not paying the first month bill in time. Find P(E_1), P(E_2). (ii) Let A denotes the event of customer paying second month's bill in time, then find P(A E_1) and P(A E_2). (iii) Find the probability of customer paying second month's bill in time. (iii) Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.

(i) P(E_1)=0.7,\ P(E_2)=0.3; (ii) P(A E_1)=0.8,\ P(A E_2)=0.4; (iii) 0.68; (iii) OR 14/17

MathsCBSE Class 12 · Board exam 2024, set 65/5/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2024: question 38

Set in the CBSE Class 12 board exam of 2024, paper 65/5/1, 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 departmental store sends bills to charge its customers once a month. Past experience shows that 70% of its customers pay their first month bill in time. The store also found that the customer who pays the bill in time has the probability of 0.8 of paying in time next month and the customer who doesn't pay in time has the probability of 0.4 of paying in time the next month.

Based on the above information, answer the following questions :

(i)
Let $E_{1}$ and $E_{2}$ respectively denote the event of customer paying or not paying the first month bill in time. Find $P (E_{1})$ , $P (E_{2})$ .
1 mark
(ii)
Let A denotes the event of customer paying second month's bill in time, then find $P (A ∣ E_{1})$ and $P (A ∣ E_{2})$ .
1 mark
(iii)
Find the probability of customer paying second month's bill in time.
2 marks
OR
(iii)
Find the probability of customer paying first month's bill in time if it is found that customer has paid the second month's bill in time.
2 marks

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 65/5/1.

Hints

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

Hint 1

$E_{1}$ and $E_{2}$ are complementary events.

Hint 2

The probabilities 0.8 and 0.4 are conditional on what happened in the first month.

Hint 3

Combine the cases with the total probability theorem; for the reverse question, use 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.

(i) 70% of customers pay the first bill in time, so $P (E_{1}) = 0.7$ and $P (E_{2}) = 1 - 0.7 = 0.3$ .
1 mark
(ii) From the given data, $P (A ∣ E_{1}) = 0.8$ and $P (A ∣ E_{2}) = 0.4$ .
1 mark
(iii) By the theorem of total probability:
$P (A) = P (E_{1}) P (A ∣ E_{1}) + P (E_{2}) P (A ∣ E_{2}) = 0.7 \times 0.8 + 0.3 \times 0.4 = 0.56 + 0.12 = 0.68 = \frac{17}{25}$ .
2 marks
(iii) OR: By Bayes' theorem:
$P (E_{1} ∣ A) = \frac{P ( E _{1} ) P ( A ∣ E _{1} )}{P ( E _{1} ) P ( A ∣ E _{1} ) + P ( E _{2} ) P ( A ∣ E _{2} )} = \frac{0.56}{0.68} = \frac{14}{17}$ .

Answer: (i) $P (E_{1}) = 0.7, P (E_{2}) = 0.3$ ; (ii) $P (A ∣ E_{1}) = 0.8, P (A ∣ E_{2}) = 0.4$ ; (iii) $0.68$ ; (iii) OR $\frac{14}{17}$

Where the marks are

$P (E_{1}) = 0.7, P (E_{2}) = 0.3$ (1 mark)
$P (A ∣ E_{1}) = 0.8, P (A ∣ E_{2}) = 0.4$ (1 mark)
Total probability gives $P (A) = 0.68$ (2 marks)
OR: Bayes' theorem gives $P (E_{1} ∣ A) = 14/17$ (2 marks)

Revise the chapter

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

Practise this chapterBoard-style questions from Probability, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 65/5/1: every question, by section.

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CBSE Class 12 Mathematics 2024: question 38

Hints

Step-by-step solution

Revise the chapter

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