Two dice are thrown simultaneously. Find the probability of getting :
(i) An even number as the sum.
(ii) The sum as a prime number.
(iii) A total of at least 10.
(iv) A doublet of even number.

Elementary events associated to the random experiment of throwing two dice are :
{ (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)}.
Total number of elementary events = 6 x 6 = 36.
(i) Let A be the event of getting an even number as the sum i.e., 2, 4, 6, 8, 10, 12.
Elementary vents favorable to event A are : {(1, 1), (1, 3), (3, 1), (2, 2), (1, 5), (5, 1), (2, 4), (4, 2), (3, 3), (2, 6), (6, 2), (4, 4), (5, 3), (3, 5), (5, 5), (6, 4), (4, 6) and (6, 6)}.
Clearly favorable number of elementary events n(A) = 18.
                   Hence, required probability P(A) =  = .
(ii) Let A be the event of getting the sum as a prime number i.e., 2, 3, 5, 7,   11.Elementary events favorable to event A are:(1, 1), (1, 2), (2, 1), (1, 4),
(4, 1), (2, 3), (3, 2), (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3), (6, 5) and (5, 6).
Favorable number of elementary events n(B)= 15.
            Hence, required probability P(B) = = .
(iii) Let A be the event of getting a total of at least 10 i.e., 10, 11, 12. Then, the elementary events favorable to A are:
(6, 4), (4, 6), (5, 5), (6, 5), (5, 6) and (6, 6).
Favorable number of elementary events n(C) = 6.
            Hence, required probability P(C) = = .
(iv) Let A be the event of getting a doublet of even number. Then, the elementary events favorable to A are
(2, 2), (4, 4) and (6, 6).
∴ Favorable number of elementary events n(D) = 3.
            Hence, required probability P(D)= = .