Two dice are thrown. The events A, B, C, D, E and F are described as follows :

A = Getting an even number on the first die.

B = Getting an odd number on the first die.

C = Getting at most 5 as sum of the numbers on the two dice.

D = Getting the sum of the numbers on the dice greater than 5 but less than 10.

E = Getting at least 10 as the sum of the numbers on the dice.

F = Getting an odd number on one of the dice.

(i) Describe the following events : A and B, B or C, B and C, A and E, A or F, A and F

(ii) State true or false :

(a) A and B are mutually exclusive.

(b) A and B are mutually exclusive and exhaustive events.

(c) A and C are mutually exclusive events.

(d) C and D are mutually exclusive and exhaustive events.

(e) C, D and E are mutually exclusive and exhaustive events.

(f) A' and B' are mutually exclusive events.

(g) A, B, F arc mutually exclusive and exhaustive events.

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Solution

When two dices are thrown, there are $6^{2} = 36$ possible outcomes.

A = Getting an even number on the number on the first die.

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 = Getting an odd number on the first die.

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 = Getting at most 5 as sum of the numbers on the two dice.

C= {(1, 1), (1,2), (1,3),(1,4),
(2,1), (2,2), (2, 3), (3,1), (3,2),(4,1)}

D = Getting the sum of the numbers on the dice > 5 but < 10

D = {(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), (5,1),(5,2), (5,3), (5,4),
(6,1),(6,2),(6,3)}

E = Getting at least 10 as the sum of the numbers on the dice.

E= {(4,6), (5,5), (5,6), (6,4),(6,5), (6,6)}

F = Getting an odd number on one of the dice.

F= {(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),
(2,1), (2,2) (2,3), (2,5),
(4,1), (4,3), (4,5),
(6,1),(6,3),(6,5)}

Its clear that A and B are mutually exclusive vents and $A \cap B = ϕ$

$B \cup C$ = {(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),
(2,1),(2,2),(2,3),(4,1)}
$B \cap C$ = {(1, 1),(1,2),(1,3),(1,4),(3,1), (3,2)}

$A \cap E$ = {(4,6), (6,4),(6,5),(6,6)}

$A \cup F$ = (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),

$A \cap F$ {(2,1),(2,3),(2,5),(4,1),(4,3), (4,5), (6,1).(6.3),(6.5)}

(ii) (a) True, because $A \cap B = ϕ$

(b) True, because $A \cap B = ϕ$ and $A \cup B = S$

(c) False, because $A \cap C \neq ϕ$

(d) False, because $A \cap B = ϕ$ and $A \cup B \neq S$

(e) True, because $C \cap D = D \cap E = C \cap E = ϕ$ and $C \cup D \cup E = S$

(f) True, because $A^{1} \cup B^{1} = ϕ$

(g) False, $A \cap F \neq ϕ$

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