Practicing the AP Board Solutions Class 11 Maths and Chapter 14 Inter 1st Year Maths Probability Solutions Exercise 14a Pdf Download will help students to clear their doubts quickly.
Intermediate 1st Year Maths Probability Solutions Exercise 14a
Probability Exercise 14a Solutions
Probability Class 11 Exercise 14a Solutions – Probability 14a Exercise Solutions
I.
Question 1.
A die is rolled. Let E be the event “die shows 4” and F be the event “die shows an even number”. Are E and F mutually exclusive?
Solution:
When a die is rolled, the sample space is given by S = {1, 2, 3, 4, 5, 6}
Accordingly, E = {4} and F = {2, 4, 6}
It is observed that E ∩ F = {4} ≠ φ
Therefore, E and F are not mutually exclusive events.
Question 2.
In the experiment of throwing a die, consider the following events.
A = {1, 3, 5}, B = {2, 4, 6}, C = {1, 2, 3}
Are these events equally likely?
Solution:
Yes, clearly A, B, and C are equally likely.
Question 3.
In the experiment of throwing a die, consider the following events.
A = {1, 3, 5}, B = {2, 4}, C = {6}
Are these events mutually exclusive?
Solution:
Yes, clearly A, B, and C are mutually exclusive.
Question 4.
In the experiment of throwing a die, consider the events
A = {2, 4, 6}, B = {3, 6}, C = {1, 5, 6}
Are these events exhaustive?
Solution:
Yes, clearly A, B, and C are exhaustive.
II.
Question 1.
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:
There are 36 elements in the sample space S = {(x, y): x, y = 1, 2, 3, 4, 5, 6}
Then A = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), (6, 6)}
B = {(1, 2), (2, 1), (1, 5), (5, 1), (3, 3), (2, 4), (4, 2), (3, 6), (6, 3), (4, 5), (5, 4), (6, 6)}
C = {(1, 1), (2, 1), (1, 2)}
D = {(6, 6)}
Now A ∩ B = {(1, 5), (2, 4), (3, 3), (4, 2), (5, 1), (6, 6)} ≠ φ
Therefore, A and B are not mutually exclusive events.
Similarly, A ∩ C ≠ φ, A ∩ D ≠ φ, B ∩ C ≠ φ, and B ∩ D ≠ φ.
Thus, the pairs of events, (A, C), (A, D), (B, C), (B, D) are not mutually exclusive events.
Further, C ∩ D ≠ φ and So C and D are mutually exclusive events.
Question 2.
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:
Three coins tossed once,
The sample space is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
A: “Three heads show.”
A = {HHH}
B: “Two heads and one tail show.”
C = {HHT, HTH, THH}
C: “Three tails show.”
C = {TTT}
D: “a head shows on the first coin.”
D = {HHH, HHT, HTH, HIT}
Mutually exclusive events
(i) (A, B), (A, C), (B, C), (C, D) (Since there are no common outcomes)
(ii) Simple events A = {HHH} and C = {TTT}
(iii) Compound events
B = {HHT, HTH, THH}
D = {HHH, HHT, HTH, HIT}
Question 3.
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 and
C: getting the sum of numbers on the dice ≤ 5.
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’, and C are mutually exclusive and exhaustive.
Solution:
A = {(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 = {(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 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
(i) It is observed that A ∩ B ≠ φ
∴ A and B are mutually exclusive.
Thus, the given statement is true.
(ii) It is observed that A ∩ B = φ and A ∪ B = S’
∴ A and B are mutually exclusive and exhaustive.
Thus, the given statement is true.
(iii) It is observed that 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)} = A
Thus, the given statement is true.
(iv) It is observed that A ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠ φ
∴ A and C are not mutually exclusive.
Thus, the given statement is false.
(v) A ∩ B’
A ∩ A = A
∴ A ∩ B’ ≠ φ
∴ A and B’ are not mutually exclusive.
Thus, the given statement is false.
(vi) It is observed that A’ ∪ B’ ∪ C = S
However, B’ ∩ C = {(2, 1), (2, 2), (2, 3), (4, 1)} ≠ φ
Therefore, events A’, B’, and C are not mutually exclusive and exhaustive.
Thus, the given statement is false.
III.
Question 1.
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’ and F’.
Solution:
S = {1, 2, 3, 4, 5, 6}
(i) A: Number less than 7 = {1, 2, 3, 4, 5, 6}
(ii) B: Number greater than 7 = { }
(iii) C: 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}
A ∩ B = { }
B ∪ C = {3, 6}
E ∩ F = {6}
D ∩ E = { }
D – E = {1, 2, 3}
A – C = {1, 2, 4, 5}
E ∩ F = {6}
(E ∩ F)c = S – {6} = {1, 2, 3, 4, 5}
Fc = S – F = {1, 2}
Question 2.
Three coins are tossed. Describe
(i) Two mutually exclusive events.
(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:
S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}
(i) Mutually exclusive = {HHH} and {TTT}
(ii) Mutually exclusive and exhaustive = {HHH}, {TTT}, and {HHT, HTH, THH, HTT, THT, TTH}
(iii) No mutually exclussive = {HHH, HHT, HTH, HIT} and {HHT, HTH, THH}
(iv) Mutually exclusive but not exhaustive = {HHH} and {TIT}
(v) Three events mutually exclusive but not exhaustive = {HHH}, {TTT} and {HTT, THT, TTH}
Question 3.
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 and
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:
When two dice are thrown, the sample space is given by
S = {(x, y): x, y = 1, 2, 3, 4, 5, 6} = {(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), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}
Accordingly, A = {(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 = {(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 = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1)}
(i) A’ = {(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, 8)} = B
(ii) Not B = 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)} = A
(iii) A or B = A ∪ B = {(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)} = S
(iv) A and B = A ∩ B = φ
(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) C’ = {(1, 5), (1, 6), (2, 4), (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’ = 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)}