Two different dice are thrown simultaneously. Find the probability of getting :

(i) sum 7

(ii) sum ≤ 3

(iii) sum ≤ 10

When two different dice are rolled together, the total number of outcomes is 6 × 6 i.e. 36 and all outcomes are equally likely. The sample space of the random experiment has 36 equally likely outcomes. The sample of the experiment

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).}

It consists of 36 equally likely outcomes.

(i) Let A be the event of getting a sum of 7,

A = {(1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)}

∴ The number of outcomes favourable to event A = 6.

∴ P(sum of 7) = $\dfrac{6}{36} = \dfrac{1}{6}.$

Hence, the probability of getting a sum of 7 is $\dfrac{1}{6}$.

(ii) Let B be the event of getting a sum ≤ 3,

B = {(1, 1), (1, 2), (2, 1)}

∴ The number of outcomes favourable to event B = 3.

∴ P(sum ≤ 3) = $\dfrac{3}{36} = \dfrac{1}{12}.$

Hence, the probability of getting a sum ≤ 3 is $\dfrac{1}{12}$.

(iii) Let C be the event of getting a sum ≤ 10,

C = {(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), (6, 1), (6, 2), (6, 3), (6, 4)}

∴ The number of outcomes favourable to event C = 33.

∴ P(sum ≤ 3) = $\dfrac{33}{36} = \dfrac{11}{12}.$

Hence, the probability of getting a sum ≤ 10 is $\dfrac{11}{12}$.