MathsCBSE Class 12 · Board exam 2026, set 65/5/1 · 3 marks · Short answer

CBSE Class 12 Mathematics 2026: question 31(b)

Set in the CBSE Class 12 board exam of 2026, paper 65/5/1, section C, question 31(b), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31(b) · 3 marks · Short answerProbability

A box contains 6 cards numbered 1 to 6. A student is asked to pick up two cards, one by one after replacement and note down the numbers on the cards. Let A be the event of getting sum of the numbers on two cards as 10, and B, the event of a number other than 4 on the first card selected.

Find P(A and B) and find whether the events A and B are independent events or not.

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

With replacement there are 36 ordered pairs. List the pairs whose sum is 10.

Hint 2

Count the outcomes whose first number is not 4.

Hint 3

Check whether $P (A \cap B) = P (A) \cdot P (B)$ .

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.

The sample space has $6 \times 6 = 36$ outcomes. $A = {(4, 6), (5, 5), (6, 4)}$ , so $n (A) = 3$ .
½ mark
B: the first card is any of 1, 2, 3, 5, 6 and the second is any of 6 numbers, so $n (B) = 5 \times 6 = 30$ .
½ mark
$A \cap B = {(5, 5), (6, 4)}$ , so $P (A \cap B) = \frac{2}{36} = \frac{1}{18}$
½ mark
$P (A) = \frac{3}{36} = \frac{1}{12}$
½ mark
$P (B) = \frac{30}{36} = \frac{5}{6}$ , so $P (A) P (B) = \frac{1}{12} \times \frac{5}{6} = \frac{5}{72} \neq = \frac{1}{18} = P (A \cap B)$
½ mark
Hence A and B are not independent events.
½ mark

Answer: $P (A and B) = \frac{1}{18}$ ; A and B are not independent

Where the marks are

Lists A, $n (A) = 3$ (½ mark)
$n (B) = 30$ (½ mark)
$n (A \cap B) = 2$ , $P (A \cap B) = \frac{2}{36}$ (½ mark)
$P (A) = \frac{1}{12}$ (½ mark)
$P (B) = \frac{5}{6}$ and $P (A) P (B) = \frac{5}{72} \neq = P (A \cap B)$ (½ mark)
Concludes not independent (½ mark)

Revise the chapter

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

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 31(b)

Hints

Step-by-step solution

Revise the chapter

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