MathsCBSE Class 12 · Sample paper, session 2024-25 · 3 marks · Short answer

CBSE Class 12 Mathematics sample paper 2024-25: question 31(a)

From the official CBSE sample question paper for 2024-25, 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 answerCompetency-basedProbability

The probability that it rains today is 0.4. If it rains today, the probability that it will rain tomorrow is 0.8. If it does not rain today, the probability that it will rain tomorrow is 0.7. If
$P_{1}$ : denotes the probability that it does not rain today.
$P_{2}$ : denotes the probability that it will not rain tomorrow, if it rains today.
$P_{3}$ : denotes the probability that it will rain tomorrow, if it does not rain today.
$P_{4}$ : denotes the probability that it will not rain tomorrow, if it does not rain today.
(i) Find the value of $P_{1} \times P_{4} - P_{2} \times P_{3}$ . [2 Marks]
(ii) Calculate the probability of raining tomorrow. [1 Mark]

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

Use complements: 'not rain' probability is 1 minus 'rain' probability in each case.

Hint 2

For (ii), rain tomorrow can happen after a rainy day or after a dry day — add both routes.

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.

$P_{1} = 1 - 0.4 = 0.6$ ; $P_{2} = 1 - 0.8 = 0.2$ ; $P_{3} = 0.7$ ; $P_{4} = 1 - 0.7 = 0.3$ .
1 mark
(i) $P_{1} \times P_{4} - P_{2} \times P_{3} = 0.6 \times 0.3 - 0.2 \times 0.7 = 0.18 - 0.14 = 0.04$ .
1 mark
(ii) Let $E_{1}$ : rains today, $E_{2}$ : does not rain today, $A$ : rains tomorrow. $P (E_{1}) = 0.4$ , $P (E_{2}) = 0.6$ , $P (A ∣ E_{1}) = 0.8$ , $P (A ∣ E_{2}) = 0.7$ .
½ mark
By total probability, $P (A) = 0.4 \times 0.8 + 0.6 \times 0.7 = 0.32 + 0.42 = 0.74$ .
½ mark

Answer: (i) 0.04; (ii) 0.74

Where the marks are

Finds $P_{1} = 0.6$ , $P_{2} = 0.2$ , $P_{3} = 0.7$ , $P_{4} = 0.3$ (1 mark)
Computes $P_{1} P_{4} - P_{2} P_{3} = 0.04$ (1 mark)
Defines events and conditional probabilities (½ mark)
Total probability gives 0.74 (½ mark)

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 paperSample paper 2024-25: every question, by section.

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CBSE Class 12 Mathematics sample paper 2024-25: question 31(a)

Hints

Step-by-step solution

Revise the chapter

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