Exercise 13.2 answers: Probability

Open NCERT p. 421Matches NCERT’s answer

1. Independent events: P(A∩B) = P(A) × P(B).
2. P(A∩B) = 3/5 × 1/5 = 3/25.

AnswerP(A∩B) = 3/25

Watch this explained “The same idea as a product”, 2:55 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 421Matches NCERT’s answer

1. 26 of the 52 cards are black. P(first card black) = 26/52.
2. After removing one black card, 25 of the remaining 51 cards are black.
3. P(both black) = 26/52 × 25/51 = 25/102.

Answer25/102

Watch this explained “Two to practise on”, 12:06 into Multiplying along a chain of dependent draws, and extending it past two events

Open NCERT p. 421Matches NCERT’s answer

1. First orange good: 12/15.
2. Second orange good (one good one already removed): 11/14.
3. Third orange good (two good ones already removed): 10/13.
4. P(all three good) = 12/15 × 11/14 × 10/13 = 44/91.

Answer44/91

Watch this explained “Two to practise on”, 12:06 into Multiplying along a chain of dependent draws, and extending it past two events

Open NCERT p. 421Matches NCERT’s answer

1. Tossing a coin and a die together gives 2 × 6 = 12 equally likely outcomes: H1, H2, …, H6, T1, T2, …, T6.
2. A (head on the coin) has 6 of these outcomes, so P(A) = 6/12 = 1/2.
3. B (3 on the die) = {H3, T3}, so P(B) = 2/12 = 1/6.
4. A∩B = {H3}, so P(A∩B) = 1/12.
5. P(A) × P(B) = 1/2 × 1/6 = 1/12, which equals P(A∩B). So A and B are independent.

AnswerYes, A and B are independent: P(A∩B) = 1/12 = 1/2 × 1/6 = P(A) × P(B).

Watch this explained “Three tests, run all the way through”, 9:37 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 421Matches NCERT’s answer

1. Even numbers on the die are 2, 4, 6, so A has 3 of the 6 outcomes: P(A) = 1/2.
2. Red numbers are 1, 2, 3, so B has 3 of the 6 outcomes: P(B) = 1/2.
3. A and B share only the outcome 2, so P(A∩B) = 1/6.
4. Independent would need P(A)×P(B) = P(A∩B). Here 1/2 × 1/2 = 1/4, which is not 1/6.

AnswerNo, A and B are not independent.

Watch this explained “Three tests, run all the way through”, 9:37 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 421Matches NCERT’s answer

1. If E and F were independent, P(E∩F) would equal P(E) × P(F).
2. P(E) × P(F) = 3/5 × 3/10 = 9/50.
3. The given P(E∩F) = 1/5 = 10/50, which is not 9/50.
4. So E and F are not independent.

AnswerNo, E and F are not independent.

Watch this explained “Dependent is just: not that”, 4:26 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

(i) mutually exclusive

1. Mutually exclusive events cannot overlap, so P(A∪B) = P(A) + P(B).
2. 3/5 = 1/2 + p, so p = 3/5 − 1/2 = 1/10.

Answerp = 1/10

(ii) independent

1. For independent events, P(A∪B) = P(A) + P(B) − P(A)×P(B).
2. 3/5 = 1/2 + p − (1/2)p = 1/2 + p/2.
3. p/2 = 3/5 − 1/2 = 1/10, so p = 1/5.

Answerp = 1/5

Watch this explained “One question, two answers”, 8:27 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

(i) P(A∩B)

1. Independent events multiply: P(A∩B) = P(A) × P(B).
2. 0.3 × 0.4 = 0.12 = 3/25.

Answer3/25 (0.12)

(ii) P(A∪B)

1. P(A∪B) = P(A) + P(B) − P(A∩B).
2. 0.3 + 0.4 − 0.12 = 0.58 = 29/50.

Answer29/50 (0.58)

(iii) P(A|B)

1. Since A and B are independent, knowing B does not change A's chance.
2. So P(A|B) = P(A) = 3/10.

Answer3/10

(iv) P(B|A)

1. Likewise P(B|A) = P(B) for independent events.
2. So P(B|A) = 2/5.

Answer2/5

Watch this explained “The same idea as a product”, 2:55 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

1. Not A and not B is the complement of A∪B.
2. P(A∪B) = P(A) + P(B) − P(A∩B) = 1/4 + 1/2 − 1/8 = 5/8.
3. P(not A and not B) = 1 − 5/8 = 3/8.

Answer3/8

Watch this explained “At least one of the two”, 13:08 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

1. Not A or not B is the complement of A∩B, so P(A∩B) = 1 − P(not A or not B) = 1 − 1/4 = 3/4.
2. Check the product: P(A) × P(B) = 1/2 × 7/12 = 7/24.
3. 7/24 is not equal to 3/4, so A and B are not independent.

AnswerNo, A and B are not independent.

Watch this explained “Three numbers that cannot happen”, 16:19 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

(i) P(A and B)

1. Independent events multiply: P(A and B) = P(A) × P(B).
2. 0.3 × 0.6 = 0.18 = 9/50.

Answer9/50 (0.18)

(ii) P(A and not B)

1. P(A and not B) = P(A) × P(not B), since independence carries over to the complement.
2. P(not B) = 1 − 0.6 = 0.4, so 0.3 × 0.4 = 0.12 = 3/25.

Answer3/25 (0.12)

(iii) P(A or B)

1. P(A or B) = P(A) + P(B) − P(A and B).
2. 0.3 + 0.6 − 0.18 = 0.72 = 18/25.

Answer18/25 (0.72)

(iv) P(neither A nor B)

1. Neither happening is the complement of A or B.
2. P(not A) × P(not B) = 0.7 × 0.4 = 0.28 = 7/25.

Answer7/25 (0.28)

Watch this explained “Complementing does not break it”, 11:16 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

1. On one throw, the chance of an odd number is 1/2, so the chance of an even number is also 1/2.
2. 'At least one odd' is everything except 'all three even'.
3. All three even, throw by throw, is (1/2)³ = 1/8.
4. So at least one odd = 1 − 1/8 = 7/8.

Answer7/8

Watch this explained “At least one of the two”, 13:08 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

(i) both balls are red

1. With replacement, the box is the same for both draws: 8 red out of 18.
2. Both red: 8/18 × 8/18 = 16/81.

Answer16/81

(ii) first ball is black and second is red

1. First black: 10/18. Second red: 8/18, since the ball drawn was put back.
2. Multiply: 10/18 × 8/18 = 20/81.

Answer20/81

(iii) one of them is black and other is red

1. 'One black, one red' can happen as black-then-red or red-then-black.
2. Each order has chance 10/18 × 8/18 = 20/81.
3. Add the two orders: 2 × 20/81 = 40/81.

Answer40/81

Watch this explained “When nothing changes”, 8:09 into Multiplying along a chain of dependent draws, and extending it past two events

Open NCERT p. 422Matches NCERT’s answer

(i) the problem is solved

1. The problem stays unsolved only if both A and B fail.
2. A fails with chance 1/2, B fails with chance 2/3: both fail = 1/2 × 2/3 = 1/3.
3. Solved = 1 − 1/3 = 2/3.

Answer2/3

(ii) exactly one of them solves the problem

1. Exactly one solving it means A solves and B fails, or B solves and A fails.
2. A solves, B fails: 1/2 × 2/3 = 1/3.
3. B solves, A fails: 1/3 × 1/2 = 1/6.
4. Add the two cases: 1/3 + 1/6 = 1/2.

Answer1/2

Watch this explained “At least one of the two”, 13:08 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 422Matches NCERT’s answer

(i) the card drawn is a spade … the card drawn is an ace

1. P(spade) = 13/52 = 1/4. P(ace) = 4/52 = 1/13.
2. The only spade ace is the ace of spades: P(E∩F) = 1/52.
3. 1/4 × 1/13 = 1/52, which matches — independent.

AnswerYes, independent

(ii) the card drawn is black … the card drawn is a king

1. P(black) = 26/52 = 1/2. P(king) = 4/52 = 1/13.
2. Black kings are the king of spades and king of clubs: P(E∩F) = 2/52 = 1/26.
3. 1/2 × 1/13 = 1/26, which matches — independent.

AnswerYes, independent

(iii) the card drawn is a king or queen … a queen or jack

1. P(king or queen) = 8/52 = 2/13. P(queen or jack) = 8/52 = 2/13.
2. The two sets share only the queens: P(E∩F) = 4/52 = 1/13.
3. 2/13 × 2/13 = 4/169, but 1/13 = 13/169 — no match, so dependent.

AnswerNo, dependent

Watch this explained “One card, asked about twice”, 0:00 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 423Matches NCERT’s answer

(a) reads neither Hindi nor English newspapers

1. Let Hindi = 0.6, English = 0.4, both = 0.2.
2. P(Hindi or English) = 0.6 + 0.4 − 0.2 = 0.8.
3. Neither = 1 − 0.8 = 0.2 = 1/5.

Answer1/5 (0.2)

(b) she reads Hindi newspaper, find the probability that she reads English

1. P(English | Hindi) = P(both) ÷ P(Hindi).
2. = 0.2 ÷ 0.6 = 1/3.

Answer1/3

(c) she reads English newspaper, find the probability that she reads Hindi

1. P(Hindi | English) = P(both) ÷ P(English).
2. = 0.2 ÷ 0.4 = 1/2.

Answer1/2

Watch this explained “Turn it round”, 7:01 into Being told one event happened shrinks the sample space you count against

Open NCERT p. 423Checked by computer

1. The only even prime number is 2 — every other even number has a factor of 2 besides itself and 1.
2. On one die, the chance of showing 2 is 1/6.
3. The two dice are independent, so both showing 2 is 1/6 × 1/6 = 1/36.

Answer1/36 (option D)

Watch this explained “Three tests, run all the way through”, 9:37 into Independence, and why it is a different idea from having no outcome in common

Open NCERT p. 423Matches NCERT’s answer

1. P(A′∩B′) always equals 1 − P(A) − P(B) + P(A∩B), whatever A and B are.
2. Option (B) says this equals [1−P(A)][1−P(B)] = 1 − P(A) − P(B) + P(A)P(B).
3. Matching the two expressions forces P(A∩B) = P(A)×P(B), which is exactly independence.
4. Mutually exclusive events (A) are actually always dependent once both have positive probability, and (C), (D) have no such link.

Answer(B) — P(A′B′) = [1 − P(A)][1 − P(B)]

Watch this explained “Complementing does not break it”, 11:16 into Independence, and why it is a different idea from having no outcome in common