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

CBSE Class 12 Mathematics 2026: question 31(b)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, 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

The probability of simultaneous occurrence of atleast one of the two events X and Y is a. If the probability that exactly one of the events X, Y occurs is b, prove that $P (X^{'}) + P (Y^{'}) = 2 - 2 a + b$ .

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

'At least one' means the union; 'exactly one' means union minus intersection.

Hint 2

Use $P (X) + P (Y) = P (X \cup Y) + P (X \cap Y)$ .

Hint 3

Write $P (X^{'}) = 1 - P (X)$ and similarly for Y.

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.

At least one of X, Y occurs: $P (X \cup Y) = a$
½ mark
Exactly one occurs: $P (X \cup Y) - P (X \cap Y) = b \Rightarrow P (X \cap Y) = a - b$
1 mark
$P (X^{'}) + P (Y^{'}) = [1 - P (X)] + [1 - P (Y)] = 2 - [P (X) + P (Y)] = 2 - [P (X \cup Y) + P (X \cap Y)]$
1 mark
$= 2 - [a + (a - b)] = 2 - 2 a + b$ . Hence proved.
½ mark

Answer: $P (X^{'}) + P (Y^{'}) = 2 - 2 a + b$ (proved)

Where the marks are

Writes $P (X \cup Y) = a$ (½ mark)
Obtains $P (X \cap Y) = a - b$ (1 mark)
Uses complements and addition theorem (1 mark)
Simplifies to $2 - 2 a + b$ (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(b)

Hints

Step-by-step solution

Revise the chapter

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