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NCERT Solutions · Class 12 · Chapter 13: Probability

NCERT Solutions for Class 12 Maths Chapter 13 Miscellaneous Exercise

The Miscellaneous Exercise on Probability. Mixed practice with conditional probability, independence and Bayes' theorem; list small sample spaces completely.

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Miscellaneous Exercise questions and solutions

Miscellaneous Exercise, Question 1

\(P(A) \ne 0\). Find \(P(B|A)\) if:
(i) A is a subset of B
Show solution
  1. \(A \cap B = A\): \(\dfrac{P(A)}{P(A)}\).
Answer: 1
(ii) \(A \cap B = \emptyset\)
Show solution
  1. \(P(A \cap B) = 0\).
Answer: 0

Practise this: Step 1, Secure the basics →

Miscellaneous Exercise, Question 2

A couple has two children.
(i) Find the probability both are boys, given at least one is a boy.
Show solution
  1. MM, MF, FM: one of three.
Answer: \(\tfrac13\)
(ii) Find the probability both are girls, given the elder is a girl.
Show solution
  1. (elder, younger): FM, FF.
Answer: \(\tfrac12\)

Practise this: Step 2, Board standard →

Miscellaneous Exercise, Question 3

5% of men and 0.25% of women have grey hair; there are equal numbers of men and women. A grey-haired person is chosen. Find the probability it is a man.
Show solution
  1. \[\begin{aligned}\dfrac{\frac12 \times 0.05}{\frac12 \times 0.05 + \frac12 \times 0.0025} &= \dfrac{0.05}{0.0525} \\ &= \dfrac{500}{525}\end{aligned}\]
Answer: \(\tfrac{20}{21}\)

Practise this: Step 2, Board standard →

Miscellaneous Exercise, Question 4

90% of people are right-handed. Find the probability that at most 6 of a random sample of 10 are right-handed.
Show solution
  1. Each person independently is right-handed with probability \(\tfrac{9}{10}\); the number of right-handers X among 10 has \[\begin{aligned}P(X &= r) \\ &= {}^{10}C_r\left(\tfrac{9}{10}\right)^r\left(\tfrac{1}{10}\right)^{10 - r}\end{aligned}\]
  2. \(P(X \le 6) = 1 - P(X \ge 7)\).
Answer: \[1 - \displaystyle\sum_{r=7}^{10}{}^{10}C_r\left(\tfrac{9}{10}\right)^r\left(\tfrac{1}{10}\right)^{10 - r}\]

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Miscellaneous Exercise, Question 5

A leap year is chosen at random. Find the probability it has 53 Tuesdays.
Show solution
  1. 366 days = 52 weeks + 2 days; the extra pair is one of 7 equally likely (Sun–Mon, Mon–Tue, …, Sat–Sun).
  2. Two of these contain a Tuesday.
Answer: \(\tfrac27\)

Practise this: Step 2, Board standard →

Miscellaneous Exercise, Question 6

Boxes A, B, C, D hold (red, white, black) marbles: A (1, 6, 3), B (6, 2, 2), C (8, 1, 1), D (0, 6, 4). A box is chosen at random and a red marble drawn.
Find the probability it came from box A, box B and box C.
Show solution
  1. Each box has 10 marbles: \[P(\text{red}|A, B, C, D) = \tfrac{1}{10}, \tfrac{6}{10}, \tfrac{8}{10}, 0\], each box \(\tfrac14\).
  2. Bayes: divide each by the total \(1 + 6 + 8 + 0 = 15\) (in tenths).
Answer: A: \(\tfrac{1}{15}\), B: \(\tfrac25\), C: \(\tfrac{8}{15}\)

Practise this: Step 2, Board standard →

Miscellaneous Exercise, Question 7

Heart attack chance is 40%; meditation and yoga reduce it by 30%, a drug by 25%. A patient picks either option with equal probability and later has a heart attack. Find the probability they followed meditation and yoga.
Show solution
  1. With yoga: \(0.4 \times 0.7 = 0.28\); with the drug: \(0.4 \times 0.75 = 0.30\).
  2. \[\dfrac{\frac12 \times 0.28}{\frac12 \times 0.28 + \frac12 \times 0.30} = \dfrac{28}{58}\]
Answer: \(\tfrac{14}{29}\)

Practise this: Step 3, Full marks on long answers →

Miscellaneous Exercise, Question 8

Each entry of a \(2 \times 2\) determinant is 0 or 1 with probability \(\tfrac12\), independently. Find the probability the determinant is positive.
Show solution
  1. 16 equally likely matrices. \(ad - bc > 0\) needs \(ad = 1\), \(bc = 0\): \(a = d = 1\) and \((b, c) \ne (1, 1)\), 3 cases.
Answer: \(\tfrac{3}{16}\)

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Miscellaneous Exercise, Question 9

P(A fails) = 0.2, P(B fails alone) = 0.15, P(A and B fail) = 0.15. Find:
(i) P(A fails | B has failed)
Show solution
  1. P(B fails) = 0.15 + 0.15 = 0.3; \(\dfrac{0.15}{0.3}\).
Answer: \(\tfrac12\)
(ii) P(A fails alone)
Show solution
  1. \(0.2 - 0.15\).
Answer: \(0.05\)

Practise this: Step 2, Board standard →

Miscellaneous Exercise, Question 10

Bag I: 3 red, 4 black; bag II: 4 red, 5 black. A ball moves from bag I to bag II, then a ball drawn from bag II is red. Find the probability the transferred ball was black.
Show solution
  1. Black moved (\(\tfrac47\)): bag II has 4 red of 10. Red moved (\(\tfrac37\)): 5 red of 10.
  2. \[\dfrac{\frac47 \cdot \frac{4}{10}}{\frac47 \cdot \frac{4}{10} + \frac37 \cdot \frac{5}{10}} = \dfrac{16}{16 + 15}\]
Answer: \(\tfrac{16}{31}\)

Practise this: Step 3, Full marks on long answers →

Miscellaneous Exercise, Question 11

\(P(A) \ne 0\) and \(P(B|A) = 1\). Then: (A) \(A \subset B\) (B) \(B \subset A\) (C) \(B = \emptyset\) (D) \(A = \emptyset\)
Show solution
  1. \(P(A \cap B) = P(A)\): whenever A happens, B happens.
Answer: (A)

Practise this: Step 1, Secure the basics →

Miscellaneous Exercise, Question 12

If \(P(A|B) > P(A)\), then: (A) \(P(B|A) < P(B)\) (B) \(P(A \cap B) < P(A)P(B)\) (C) \(P(B|A) > P(B)\) (D) \(P(B|A) = P(B)\)
Show solution
  1. \(P(A \cap B) > P(A)P(B)\); dividing by \(P(A)\): \(P(B|A) > P(B)\).
Answer: (C)

Practise this: Step 2, Board standard →

Miscellaneous Exercise, Question 13

\(P(A) + P(B) - P(A \text{ and } B) = P(A)\). Then: (A) \(P(B|A) = 1\) (B) \(P(A|B) = 1\) (C) \(P(B|A) = 0\) (D) \(P(A|B) = 0\)
Show solution
  1. So \(P(B) = P(A \cap B)\), and \(P(A|B) = \dfrac{P(A \cap B)}{P(B)} = 1\).
Answer: (B)

Practise this: Step 2, Board standard →

Done the NCERT exercises? The board paper asks more

Probability has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.

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Probability in our sample papers: Sample paper 1 (questions 18, 20, 25, 36) · Sample paper 2 (questions 20, 31, 38) · Sample paper 3 (questions 18, 20, 25, 36) · Sample paper 4 (questions 20, 31, 38) · Sample paper 5 (questions 18, 20, 25, 36).

Also useful: free MCQs and case studies for Probability · formulas for this chapter (Class 12 formula sheet, free PDF) · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan

Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.