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 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) (iv) A and C are mutually exclusive (v) A and are mutually exclusive (vi) are mutually exclusive and exhaustive.
Let the event A: getting an even 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 ) }
Let the event 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 ) }
Let the event C: getting the sum of the numbers on the dice is less than or equal to5.
C = { ( 1 , 1 ) , ( 1 , 2 ) , ( 1 , 3 ) , ( 1 , 4 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 4 , 1 ) }
(i)
Two events, A and B, are mutually exclusive if, their intersection is a null set, A ∩ B = ϕ .
It can be observed that,
A ∩ B = ϕ
Thus, the given statement is true.
(ii)
Two events, A and B, are exhaustive if, their union gives the complete sample space, A ∪ B = S .
It can be observed that,
A ∩ B = ϕ A ∪ B = S
Events A and B are both mutually exclusive and exhaustive. Thus, the given statement is true.
(iii)
It can be 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)
Solve.
A ∩ C= [ { ( 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 ) , ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) , ( 3 , 2 ) , ( 4 , 1 ) } ] = { ( 2 , 1 ) , ( 2 , 2 ) , ( 2 , 3 ) , ( 4 , 1 ) } ≠ ϕ
Thus, the given statement is false.
(v)
Two events, A and B, are mutually exclusive if, their intersection is a null set, A ∩ B = ϕ .
A ∩ B ′ = A ∩ A = A ≠ ϕ
Both events A and B ′ are not mutually exclusive. Thus, the given statement is false.
(vi)
Three events, A, B and C are exhaustive if, their union gives the complete sample space, A ∪ B ∪ C = S
Three events, A, B and C are mutually exclusive if, their intersection gives a null set, A ∩ B ∩ C = ϕ
A ′ ∩ B ′ = B ∩ B ′ = ϕ
A ′ ∩ B ′ ∩ C = ϕ ∩ C = ϕ
Hence, A, B and C are mutually exclusive.
A ′ ∪ B ′ = B ∪ B ′ = S
A ′ ∪ B ′ ∪ C = S ∪ C = S
Hence, A, B and C are mutually exclusive.
Events A, B and C are both mutually exclusive and exhaustive. Thus, the statement is true.
Let the event A: getting an even number on the first die.
Let the event B: getting an odd number on the first die.
Let the event C: getting the sum of the numbers on the dice is less than or equal to5.
(i)
Two events, A and B, are mutually exclusive if, their intersection is a null set,
It can be observed that,
Thus, the given statement is true.
(ii)
Two events, A and B, are exhaustive if, their union gives the complete sample space,
It can be observed that,
Events A and B are both mutually exclusive and exhaustive. Thus, the given statement is true.
(iii)
It can be observed that,
Thus, the given statement is true.
(iv)
Solve.
Thus, the given statement is false.
(v)
Two events, A and B, are mutually exclusive if, their intersection is a null set,
Both events A and
(vi)
Three events, A, B and C are exhaustive if, their union gives the complete sample space,
Three events, A, B and C are mutually exclusive if, their intersection gives a null set,
Hence, A, B and C are mutually exclusive.
Hence, A, B and C are mutually exclusive.
Events A, B and C are both mutually exclusive and exhaustive. Thus, the statement is true.
