MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 31(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section C, question 31(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(a) · 3 marks · Short answerProbability

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

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

Hints

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

Hint 1

The die decides which bag is used — define events for each bag.

Hint 2

Numbers less than 3 are 1 and 2.

Hint 3

Use the theorem of total probability.

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 $E_{1}$ : die shows a number less than 3 (bag I chosen); $E_{2}$ : die shows 3 or more (bag II chosen); $A$ : red ball is drawn.
½ mark
$P (E_{1}) = \frac{2}{6}$ , $P (A ∣ E_{1}) = \frac{3}{7}$ , $P (E_{2}) = \frac{4}{6}$ , $P (A ∣ E_{2}) = \frac{8}{14}$
1 mark
By the theorem of total probability:
$P (A) = P (E_{1}) P (A ∣ E_{1}) + P (E_{2}) P (A ∣ E_{2}) = \frac{2}{6} \cdot \frac{3}{7} + \frac{4}{6} \cdot \frac{8}{14}$
1 mark
$= \frac{1}{7} + \frac{8}{21} = \frac{3 + 8}{21} = \frac{11}{21}$
½ mark

Answer: $\frac{11}{21}$

Where the marks are

Defines events $E_{1}$ , $E_{2}$ , $A$ (½ mark)
Correct probabilities $P (E_{1}), P (E_{2}), P (A ∣ E_{1}), P (A ∣ E_{2})$ (1 mark)
Applies total probability theorem (1 mark)
Final answer $\frac{11}{21}$ (½ 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 paper2026 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2026: question 31(a)

Hints

Step-by-step solution

Revise the chapter

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