Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is :

(i) 8

(ii) 13

(iii) less than or equal to 12

When two dice are thrown simultaneously;

Number of possible outcomes = 6 × 6 = 36.

(i) For obtaining a total of 8, favourable outcomes are : {(2, 6), (3, 5), (4, 4), (5, 3), (6, 2)}.

∴ Number of favourable outcomes = 5.

P(that the sum of the two numbers appearing on the top of the dice is 8) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{5}{36}$.

Hence, the probability that the sum of the two numbers appearing on the top of the dice is 8 = $\dfrac{5}{36}$.

(ii) There is no outcome favourable to obtaining a sum of 13.

∴ Number of favourable outcomes = 0.

P(that the sum of the two numbers appearing on the top of the dice is 13) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}}$ = 0.

Hence, the probability that the sum of the two numbers appearing on the top of the dice is 13 = 0.

(iii) The sum of all the outcomes is either less than or equal to 12.

∴ Number of favourable outcomes = 36.

P(that the sum of the two numbers appearing on the top of the dice is less than or equal to 12) = $\dfrac{\text{No. of favourable outcomes}}{\text{No. of possible outcomes}} = \dfrac{36}{36}$ = 1.

Hence, the probability that the sum of the two numbers appearing on the top of the dice is less than or equal to 12 is 1.