NCERT Solutions for Class 11 Maths Chapter 14 Probability Ex 14.1
Ex 14.1 Class 11 Maths Question 1.
A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive?
Solution:
An experiment involves rolling a die.
∴ Sample space, S = {1, 2, 3, 4, 5, 6}
E: die shows 4 = {4}
F: die shows an even number = {2, 4, 6}
∴ E ∩ F = {4} ⇒ E ∩ F ≠ ⏀
Therefore, E and F are not mutually exclusive.
Ex 14.1 Class 11 Maths Question 2.
A die is thrown. Describe the following events:
(i) A: a number less than 7
(ii) B: a number greater than 7
(iii) C: a multiple of 3
(iv) D: a number less than 4
(v) E: an even number greater than 4
(vi) F: a number not less than 3
Also find A ∪ B, A ∩ B, B ∪ C, E ∩ F, D ∩ E, A – C, D – E, E ∩ F’, F’.
Solution:
An experiment involves rolling a die.
∴ Sample space, S = {1, 2, 3, 4, 5, 6}
(i) A: a number less than 7 = {1, 2, 3, 4, 5, 6}
(ii) B: a number greater than 7 = ⏀
(iii) C: a multiple of 3 = {3, 6}
(iv) D: a number less than 4 = {1, 2, 3}
(v) E: an even number greater than 4 = {6}
(vi) F: a number not less than 3 = {3, 4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6) ∪ ⏀
= {1, 2, 3, 4, 5, 6}
A ∩ B = {1, 2, 3, 4, 5, 6) ∩ ⏀ = ⏀
B ∪ C = ⏀ ∪ {3, 6} = {3, 6}
E ∩ F = {6} ∩ {3, 4, 5, 6) = {6}
D ∩ E = {1, 2, 3} ∩ (6} = ⏀
A – C = (1, 2, 3, 4, 5, 6) – {3, 6} = {1, 2, 4, 5}
D – E = {1, 2, 3} – {6} = {1, 2, 3}
F’ = {1, 2, 3, 4, 5, 6) – {3, 4, 5, 6) = {1, 2)
E ∩ F’={6} ∩ {1, 2} = ⏀
Ex 14.1 Class 11 Maths Question 3.
An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events:
A: the sum is greater than 8.
B: 2 occurs on either die.
C: the sum is at least 7 and a multiple of 3.
Which pairs of these events are mutually exclusive?
Solution:
An experiment involves rolling a pair of dice.
∴ Sample space = 6 × 6 = 62 = 36 possible outcomes.
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Now,
A: the sum is greater than 8
= {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)}
B: 2 occurs on either die = {(1, 2), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 2), (4, 2), (5, 2), (6, 2)}
C: The sum is at least 7 and a multiple of 3 = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)}
A ∩ B =⏀, B ∩ C = ⏀
Thus, the above relations show that A and B; B and C are mutually exclusive events.
Ex 14.1 Class 11 Maths Question 4.
Three coins are tossed once. Let A denote the event “three heads show”, B denote the event “two heads and one tail show”, C denote the event “three tails show” and D denote the event “a head shows on the first coin”. Which events are
(i) Mutually exclusive?
(ii) Simple?
(iii) Compound?
Solution:
An experiment involves tossing three coins:
∴ Sample space, S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
The given events are:
A: Three heads show = {HHH}
B: Two heads and one tail show = {HHT, HTH, THH}
C: Three tails show = {TTT}
D: A head shows on the first coin = {HHH, HHT, HTH, HTT}
(i) Since A ∩ B = ⏀, A ∩ C = ⏀, B ∩ C = ⏀, C ∩ D = ⏀.
⇒ A and B; A and C; B and C; C and D are mutually exclusive events.
(ii) A and C are simple events.
(iii) B and D are compound events.
Ex 14.1 Class 11 Maths Question 5.
Three coins are tossed. Describe
(i) Two events which are mutually exclusive.
(ii) Three events which are mutually exclusive and exhaustive.
(iii) Two events, which are not mutually exclusive.
(iv) Two events which are mutually exclusive but not exhaustive.
(v) Three events which are mutually exclusive but not exhaustive.
Solution:
An experiment involves tossing three coins.
The sample space, S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
(i) Two events A and B which are mutually exclusive are:
A: “getting at most one head” and B: “getting at most one tail”
(ii) Three events A, B and C which are mutually exclusive and exhaustive are:
A: “getting at least two heads”
B: “getting exact two tails” and C: “getting exactly three tails”
(iii) Two events A and B which are not mutually exclusive are:
A: “getting exactly two tails” and B: “getting at most two heads”
(iv) Two events A and B which are mutually exclusive but not exhaustive are:
A: “getting at least two heads” and B: “getting at least three tails”
(v) Three events A, B and C which are mutually exclusive but not exhaustive are:
A: “getting at least three tails”
B: “getting at least three heads”
C: “getting exactly two tails”
Ex 14.1 Class 11 Maths Question 6.
Two dice are thrown. The events A, B and C are as follows:
A: getting an even number on the first die.
B: getting an odd number on the first die.
C: getting the sum of the numbers on the dice ≤ 5.
Describe the events
(i) A’
(ii) not B
(iii) A or B
(iv) A and B
(v) A but not C
(vi) B or C
(vii) B and C
(viii) A ∩ B’ ∩ C’
Solution:
An experiment involves rolling two dice.
The sample space, S = 6 × 6 = 36 outcomes.
S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
A: getting an even number on the first die = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
B: getting an odd number on the first die = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
C: getting the sum of the numbers on the dice ≤ 5 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
(i) A’: getting an odd number on the first die = B
(ii) not B: getting an even number on the first die = A
(iii) A or B = A ∪ B = S
∴ A ∪ B = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (3, 1), (3, 2), (3, 3), (3, 4 ), (3, 5), (3, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
(iv) A and B = A ∩ B = ⏀
(v) A but not C = A – C = {(2, 4), (2, 5), (2, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
(vi) B or C = B ∪ C = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6)}
(vii) B and C = B ∩ C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)}
(viii) A: getting an even number on the first die = B’
B’: getting an even number on the first die = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
C’: getting the sum of numbers on two dice > 5 = {(1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 5) (3, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
∴ A ∩ B’∩ C’ = {(2, 4), (2, 5), (2, 6), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Ex 14.1 Class 11 Maths Question 7.
Refer to question 6 above, state true or false: (give reason for your answer).
(i) A and B are mutually exclusive.
(ii) A and B are mutually exclusive and exhaustive.
(iii) A = B’
(iv) A and C are mutually exclusive.
(v) A and B’ are mutually exclusive.
(vi) A’, B’, C are mutually exclusive and exhaustive.
Solution:
(i) True.
A = getting an even number on the first die.
B = getting an odd number on the first die.
There are no common elements in A and B.
⇒ A ∩ B = ⏀
∴ A and B are mutually exclusive.
(ii) True.
From (i), A and B are mutually exclusive.
A ∪ B = {(1, 1), (1, 2), … (1, 6), (2, 1), (2, 2), … (2, 6),…, (6, 1), (6, 2), …, (6, 6)} = S
∴ A ∪ B is mutually exhaustive.
(iii) True.
B = getting an odd number on the first die.
B’ = getting an even number on first die = A.
∴ A = B’
(iv) False.
We have, A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)}
Since A ∩ C ≠ ⏀, therefore, A and C are not mutually exclusive.
(v) False.
Since A = B’ [from (iii)]
∴ A ∩ B’= A ∩ A = A ≠ ⏀
(vi) False.
Since A’ = B and B’ = A, A’ ∩ B’ = ⏀
B ∩ C = {(1, 1), (1, 2), (1, 3), (1, 4), (3, 1), (3, 2)} ≠ ⏀
A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠ ⏀
Thus, A’, B’ and C are not mutually exclusive.
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