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Class 12 Maths Chapter 13: Probability MCQs with answers

10 multiple-choice questions on Probability, each with its answer and the reason, written from the NCERT chapter. For the CBSE board and JEE Main, which are set from the same NCERT chapter. Open a question's solution to check your working; then take the free test to find your next chapter.

Question 1
For events E and F of a sample space, P(F) = 0.4 and P(E′|F) = 0.35. Then P(E ∩ F) is
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Answer: (a) 0.26

P(E|F) = 1 − P(E′|F) = 0.65, so P(E ∩ F) = P(F) P(E|F) = 0.4 × 0.65 = 0.26. Option (b) uses 0.35 in place of 0.65; (c) stops at P(E|F); (d) subtracts.

Question 2
Galileo explained why, when three fair dice are thrown, a total of 10 turns up more often than a total of 9. If the 216 outcomes are equally likely, P(total 10) − P(total 9) equals
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Answer: (b) 1108

Counting ordered triples, 27 outcomes give a total of 10 and 25 give a total of 9. The difference is (27 − 25)/216 = 2/216 = 1/108. Option (d) comes from counting the six unordered partitions of each total, which are not equally likely.

Question 3
In a lucky draw at a mela in Varanasi, one token is drawn at random from 40 tokens numbered 1 to 40, and E is the event 'the number is even'. The announcer gives one clue about the number. After which clue is the probability of E the greatest?
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Answer: (c) P(E | the number is a perfect square)

Each clue reduces the sample space. Primes up to 40: 12, of which only 2 is even, 1/12. Multiples of 3: 13, of which the 6 multiples of 6 are even, 6/13. Perfect squares {1, 4, 9, 16, 25, 36}: 3 even, 1/2. Less than 6: {1, 2, 3, 4, 5}, 2 even, 2/5. The greatest is 1/2, after the perfect-square clue.

Question 4
A, B and F are events with P(F) ≠ 0, P(A|F) = 12, P(B|F) = 13 and P((A ∪ B)|F) = 23. Then P((A ∩ B)|F) is
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Answer: (d) 16

P((A ∪ B)|F) = P(A|F) + P(B|F) − P((A ∩ B)|F), so P((A ∩ B)|F) = 1/2 + 1/3 − 2/3 = 1/6. Option (a) is only P(A|F) + P(B|F).

Question 5
In a town in Assam, the probability that it rains on a given morning is 0.3, and the probability that the school bus is late on a rainy morning is 0.6. The probability that on a given morning it rains and the bus is late is
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Answer: (b) 0.18

With R the event 'it rains' and L 'the bus is late', P(R ∩ L) = P(R) P(L|R) = 0.3 × 0.6 = 0.18. Option (a) adds; (c) divides; (d) uses P(R′) = 0.7.

Question 6
A jar at a school fair holds n coupons, exactly 2 of which win a prize. Two coupons are drawn one after the other without replacement. If the probability that both coupons win is 115, then n is
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Answer: (c) 6

P(first wins) = 2/n and P(second wins | first wins) = 1/(n − 1), so 2/(n(n − 1)) = 1/15, n(n − 1) = 30 and n = 6. Check: 2/6 × 1/5 = 1/15. Option (a) gives 1/10; (b) treats the draws as one draw of a winner.

Question 7
A parcel from Surat to Shillong passes through three hubs. The probability that it clears the first hub on time is 0.9; given that it has cleared the first on time, the probability that it clears the second on time is 0.8; given that it has cleared both on time, the probability that it clears the third on time is 0.75. The probability that it clears all three hubs on time is
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Answer: (d) 0.54

By the multiplication rule for three events, P(E ∩ F ∩ G) = P(E) P(F|E) P(G|EF) = 0.9 × 0.8 × 0.75 = 0.54. Option (b) stops after two hubs; (c) multiplies the three complements.

Question 8
A and B are independent events with P(A) = 0.3 and P(A ∩ B) = 0.12. Then P(B′) is
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Answer: (a) 0.6

For independent events P(A ∩ B) = P(A) P(B), so P(B) = 0.12/0.3 = 0.4 and P(B′) = 1 − 0.4 = 0.6. Option (b) is P(B); (c) is 1 − P(A ∩ B).

Question 9
A and B are mutually exclusive events with P(A) = 0.3 and P(B) = 0.4. Which of the following is true?
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Answer: (c) P(A|B) = 0

Mutually exclusive means A ∩ B = φ, so P(A ∩ B) = 0 and P(A|B) = 0/0.4 = 0. Since P(A) P(B) = 0.12 ≠ 0, A and B are not independent, which rules out (a) and (b); P(A ∪ B) = 0.3 + 0.4 = 0.7, not 0.58, which would be the value for independent events.

Question 10
A spinner has 8 equal sectors numbered 1 to 8 and is spun once. E is the event 'the number is even', and G is an event made up of four of the eight numbers. E and G are independent exactly when the number of even numbers in G is
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Answer: (a) 2

P(E) = 1/2 and P(G) = 4/8 = 1/2. If G contains m even numbers, P(E ∩ G) = m/8. Independence needs m/8 = 1/2 × 1/2 = 1/4, so m = 2; for example G = {1, 2, 3, 4}. Option (b) would make G = E, with P(E ∩ G) = 1/2.

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