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?
No.
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′.
(i) A = {1, 2, 3, 4, 5, 6}
(ii) B = ∅
(iii) C = {3, 6}
(iv) D = {1, 2, 3}
(v) E = {6}
(vi) F = {3, 4, 5, 6}
A ∪ B = {1, 2, 3, 4, 5, 6}
A ∩ B = ∅
B ∪ C = {3, 6}
E ∩ F = {6}
D ∩ E = ∅
A − C = {1, 2, 4, 5}
D − E = {1, 2, 3}
E ∩ F′ = ∅
F′ = {1, 2}
An experiment involves rolling a pair of dice. Describe the 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?
A = {(3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)}
B = {(1,2), (2,2), (3,2), (4,2), (5,2), (6,2), (2,1), (2,3), (2,4), (2,5), (2,6)}
C = {(3,6), (6,3), (5,4), (4,5), (6,6)}
A and B, B and C are mutually exclusive.
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’. Determine whether the events are:
(i) mutually exclusive
(ii) simple
(iii) compound
(i) A and B; A and C; B and C; C and D are mutually exclusive.
(ii) A and C are simple.
(iii) B and D are compound.
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.
(i) “Getting at least two heads” and “getting at least two tails”
(ii) “Getting no heads”, “getting exactly one head” and “getting at least two heads”
(iii) “Getting at most two tails” and “getting exactly two tails”
(iv) “Getting exactly one head” and “getting exactly two heads”
(v) “Getting exactly one tail”, “getting exactly two tails”, “getting exactly three tails”
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′
(i) A′ = B
(ii) B′ = A
(iii) A ∪ B = S
(iv) A ∩ B = ∅
(v) 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 ∪ 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 ∩ C = {(1,1),(1,2),(1,3),(1,4),(3,1),(3,2)}
(viii) 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)}
Referring to question 6, state true or false (with reasons):
(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
(i) True
(ii) True
(iii) True
(iv) False
(v) False
(vi) False