4. Two dice are thrown find the probability of getting :
(i) The sum of the numbers on their upper faces is divisible by 9.
(ii) The sum of the numbers on their upper faces is at most 3.
(iii) The number on the upper face of the first die is less than the number on the upper face of the second die.
Sol. Two dice are thrown
∴ S = { (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
(i) Let A be the event that sum of numbers on their upper faces is divisible by 9
A = { (3, 6), (4, 5), (5, 4), (6, 3) }
n (A) = 4
P (A) = n (A)/n (S)
∴ P (A) = 4/36
∴ P (A) = 1/9
(ii) Let B be the event that sum of number on their upper faces is at the most 3.
B = { (1, 1), (1, 2), (2, 1) }
n (B) = 3
P (B) = n (B)/n (S)
∴ P (B) = 3/36
∴ P (B) = 1/12
(iii)Let C be the event that number on the upper face of the first die is less than the number on the upper face of second die.
C = { (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 3), (2, 4), (2, 5), (2, 6) (3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6) }
n (C) = 15
P (C) = n (C)/n (S)
∴ P (C) = 15/36
∴ P (C) = 5/12
