NCERT Solutions Class 11 Mathematics Probability Exercise 16.2

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?

Answer:
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} ≠ ф
Therefore, E and F are not mutually exclusive events.

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, E ∪ F, D ∩ E, A – C, D – E, F′, E ∩ F′

Answer:
When a die is thrown, the sample space is given by 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 = ф
(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}
Now,
A U B = {1, 2, 3, 4, 5, 6},              A ∩ B = ф
A U C = {3, 6},                               E ∩ F = {6}
D ∩ E = ф,                                     A – C = {1, 2, 4, 5}
D – E = {1, 2, 3},                           F’ = {1, 2}                        E ∩ F’ = ф

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 occur on either die              
C: the sum is at least 7 and a multiple of 3. Which pairs of these events are mutually exclusive?

Answer:
When a pair of dice is rolled, the sample space is given by:
S = {(x, y): x comes from 1^st dice and y comes from 2^nd dice and 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)
                (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: the sum is greater than 8,            
A = {(3, 6), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 3), (6, 4), (6, 5), (6, 6)}
B: 2 occur on either die              
B = {(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (1, 2), (3, 2), (4, 2), (5, 2), (6, 2)}
C: the sum is at least 7 and a multiple of 3
C = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)}
It is observed that
A ∩ B = ф
B ∩ C = ф
C ∩ A = {(3, 6), (4, 5), (5, 4), (6, 3), (6, 6)} ≠ ф
Hence, events A and B and events B and C are mutually exclusive.

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?

Answer:
When three coins are tossed, the sample space is given by
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Accordingly,
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}
We now observe that
A ∩ B = ф,         A ∩ C = ф,                                    A ∩ D = {HHH} ≠ ф
B ∩ C = ф,        B ∩ D = {HHT, HTH} ≠ ф            C ∩ D = ф
(i) Event A and B; event A and C; event B and C; and event C and D are all mutually exclusive.
(ii) If an event has only one sample point of a sample space, it is called a simple event.
Thus, A and C are simple events.
(iii) If an event has more than one sample point of a sample space, it is called a compound
event.
Thus, B and D are compound events.

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.

Answer:
When three coins are tossed, the sample space is given by
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
(i) Two events that are mutually exclusive can be
A: getting no heads and           B: getting no tails
This is because sets
A = {TTT} and B = {HHH} are disjoint.
(ii) Three events that are mutually exclusive and exhaustive can be
A: getting no heads       B: getting exactly one head           C: getting at least two heads
i.e. A= {TTT}
B= {HTT, THT, TTH}
C= {HHH, HHT, HTH, THH}
This is because A ∩ B = B ∩ C = C ∩ A = ф and A U B U C = S
(iii) Two events that are not mutually exclusive can be
A: getting three heads B: getting at least 2 heads
A= {HHH}
B= {HHH, HHT, HTH, THH}
This is because A ∩ B = {HHH} ≠ ф
(iv) Two events which are mutually exclusive but not exhaustive can be
A: getting exactly one head
B: getting exactly one tail
A= {HTT, THT, TTH}
B= {HHT, HTH, THH}
It is because, A ∩ B = ф, but A U B ≠ S
(v) Three events that are mutually exclusive but not exhaustive can be
A: getting exactly three heads
B: getting one head and two tails
C: getting one tail and two heads
A= {HHH}
B= {HTT, THT, TTH}
C= {HHT, HTH, THH}
This is because A ∩ B = B ∩ C = C ∩ A = ф, but A U B U C ≠ S