CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q313 Marksprobability
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Question 31 (Probability)
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. OR The probability of simultaneous occurrence of at least 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 − 2a + b.
Alternative 1 3 Marks
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.
Answer: 11/21
- Let E₁: number <3 (2/6), E₂: number ≥3 (4/6).
- P(red|E₁) = 3/7, P(red|E₂) = 8/14 = 4/7. P(red) = P(E₁)P(red|E₁) + P(E₂)P(red|E₂) = (2/6)(3/7) + (4/6)(4/7) = 6/42 + 16/42 = 22/42 = 11/21.
- 11/21
Alternative 2 3 Marks
The probability of simultaneous occurrence of at least 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 − 2a + b.
Answer: Proved.
- P(X∪Y) = a, P(exactly one) = P(X∪Y) − P(X∩Y) = b ⇒ P(X∩Y) = a−b.
- P(X′) + P(Y′) = 2 − [P(X)+P(Y)] = 2 − [P(X∪Y) + P(X∩Y)] = 2 − [a + (a−b)] = 2 − 2a + b.
- Proved.