MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2026: question 37

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedProbability

There are three types of vaccines A $_{1}$ , A $_{2}$ , A $_{3}$ , available in the market to protect the population of the country from spread of certain infection. According to a survey conducted, it was found that 25% of the population was given Vaccine A $_{1}$ , 35% of the population was given Vaccine A $_{2}$ and 40% of the population was given Vaccine A $_{3}$ . The survey also stated that the probabilities that Vaccines A $_{1}$ , A $_{2}$ and A $_{3}$ would protect against the infection were 60%, 55% and 50% respectively.

Based on the above information, answer the following questions :

Find the probability that :

(i)
The person taking vaccine A $_{2}$ will get infected.
1 mark
(ii)
If a person is chosen randomly, he/she will be protected from the infection.
1 mark
(iii)(a)
The person was given Vaccine A $_{1}$ , given that the randomly chosen person is infected.
2 marks
OR
(iii)(b)
The person was given Vaccine A $_{3}$ , given that the randomly chosen person is not infected.
2 marks

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/5/1.

Hints

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

Hint 1

If a vaccine protects with probability 55%, what is the probability of infection with it?

Hint 2

For a randomly chosen person, add P(vaccine) × P(protected | vaccine) over all three vaccines.

Hint 3

For (iii), use Bayes' theorem: the required term divided by the total probability of being infected (or not infected).

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.

Let $E_{1}, E_{2}, E_{3}$ be the events that a person was given $A_{1}, A_{2}, A_{3}$ . Let I be the event that the person is infected.
Then $P (E_{1}) = 0.25$ , $P (E_{2}) = 0.35$ , $P (E_{3}) = 0.40$ and $P (I^{'} ∣ E_{1}) = 0.60$ , $P (I^{'} ∣ E_{2}) = 0.55$ , $P (I^{'} ∣ E_{3}) = 0.50$ .
Hence $P (I ∣ E_{1}) = 0.40$ , $P (I ∣ E_{2}) = 0.45$ , $P (I ∣ E_{3}) = 0.50$ .
(i) $P (infected ∣ A_{2}) = 1 - 0.55 = 0.45$ (i.e. 45%).
1 mark
(ii) By the total probability theorem:
$P (I^{'}) = 0.25 (0.60) + 0.35 (0.55) + 0.40 (0.50)$
½ mark
(ii) $= 0.15 + 0.1925 + 0.20 = 0.5425$ .
½ mark
(iii)(a) $P (I) = 0.25 (0.40) + 0.35 (0.45) + 0.40 (0.50) = 0.1 + 0.1575 + 0.2 = 0.4575$ .
By Bayes' theorem: $P (E_{1} ∣ I) = \frac{P ( E _{1} ) P ( I ∣ E _{1} )}{P ( I )} = \frac{0.25 \times 0.40}{0.4575}$ .
1½ marks
(iii)(a) $= \frac{0.1}{0.4575} = \frac{1000}{4575} = \frac{40}{183}$ .
½ mark
(iii)(b) By Bayes' theorem: $P (E_{3} ∣ I^{'}) = \frac{P ( E _{3} ) P ( I ^{'} ∣ E _{3} )}{P ( I ^{'} )} = \frac{0.40 \times 0.50}{0.25 ( 0.60 ) + 0.35 ( 0.55 ) + 0.40 ( 0.50 )}$ .
1½ marks
(iii)(b) $= \frac{0.2}{0.5425} = \frac{2000}{5425} = \frac{80}{217}$ .
½ mark

Answer: (i) 0.45; (ii) 0.5425; (iii)(a) $\frac{40}{183}$ ; (iii)(b) $\frac{80}{217}$

Where the marks are

(i) Probability of infection with $A_{2}$ is 0.45 (1 mark)
(ii) Writes the total probability expression for protection (½ mark)
(ii) Evaluates it as 0.5425 (½ mark)
(iii)(a) Correct Bayes' theorem expression for $P (A_{1} ∣ infected)$ (1½ marks)
(iii)(a) Simplifies to $\frac{40}{183}$ (½ mark)
(iii)(b) Correct Bayes' theorem expression for $P (A_{3} ∣ not infected)$ (1½ marks)
(iii)(b) Simplifies to $\frac{80}{217}$ (½ mark)

Revise the chapter

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

Practise this chapterBoard-style questions from Probability, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/5/1: 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 2026: question 37

Hints

Step-by-step solution

Revise the chapter

More questions from Probability