Two fair dice are rolled simultaneously. Find the probability of the following events:

(1) A : getting the same number on both dice. (2) B : the sum of the integers on two dice is more than 4 but less than 8.

(3) C : the product of numbers on two dice is divisible by 2.

(4) D : the sum of numbers on two dice is greater than 12.

If two dice are rolled together then we get 36 outcomes as

(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)

(4,4),(4,2) ,(4,3),(4,4),(4,5) (4,6)

(5,5),(5,2),(5,3),(5,4),(5,5) (5,6)

(6,1),(6,2) ,(6,3),(6,4),(6,5) (6,6)

Favourable outcomes of event A i.e, getting the same number on both sides are (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)

Number of favourable outcomes of event A = 6

Favourable outcomes of event B i.e, the sum of the integers on two dice is more than 4 but less than 8 are (1,4), (1,5), (1,6), (2,3), (2,4),(2,5), (3,2),

(3,3), (3,4), (4,1), (4,2), (4,3), (5,1), (5,2), (6,1),

Number of favourable outcomes of event B = 15

Favourable outcomes of event C i.e, the product of numbers on two dice is divisible by 2 are

(1,2),(1,4),(1,6)

(2,1),(2,2) ,(2,3),(2,4), (2,5)(2,6)

(3,2),(3,4)(3,6)

(4,1),(4,2) ,(4,3),(4,4) ,(4,5),(4,6)

(5,2),(5,4)(5,6)

(6,1),(6,2) ,(6,3),(6,4),(6,5),(6,6)

Number of favourable outcomes of event C = 27

Favourable outcomes of event D i.e, the sum of numbers on two dice is greater than 12 is 0 as the greatest value of the sum of numbers on two dice is 6 + 6 = 12.

Number of favourable outcomes of event D = 0

Total number of favourable outcomes = 36

(i) Probability of occurrence of event A = P(A)

P(A) = = 0.16667

(ii) Probability of occurrence of event B = P(B)

P(B) = = 0.41667

(iii) Probability of occurrence of event C = P(C)

P(C) = = 0.75

(iv) Probability of occurrence of event D = P(D)

P(D) = = 0