Question

In each of the following experiments, write the sample space S, number of sample points n(S), event A,B,C and n(A), n(B), n(C), also find complementary events, mutually exclusive events:

Two dice are thrown, A is the event that the sum of the numbers on their upper face is at least nine, B is the event that the sum of the numbers on their upper face is divisible by 8, C is the event that the same number on the upper faces of both dice.

Answer 1

Given:
Two dice are thrown.
Thus, the sample space is {(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)}.
∴ n(S) = 36
Now, A is the event that the sum of the numbers on the upper faces of the dice is at least 9.
∴ A = {(3,6), (4,5), (4,6), (5,4), (5,5), (5,6), (6,3), (6,4), (6,5), (6,6)}
Or,
n(A) = 10
Now,
B is the event that the sum of the numbers on the upper faces of the dice is divisible by 8.
∴ B = {(2,6), (3,5), (4,4), (5,3), (6,2)}
Or,
n(B) = 5
Also,
C is the event of getting the same number on the upper faces of the dice.
∴ C = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
Or,
n(C) = 6
Here, A $\cap$ C = $\varnothing$
Events A and C are mutually exclusive.
And,
B $\cap$ C = $\varnothing$
Events B and C are mutually exclusive.