Step 1: Identify the total number of out comes and favourable outcomes.

Event: Two dies are thrown at the same time

When we tossed a die , than total number of events ( outcomes ) = 6

But when we tossed two dice simultaneously , than total number of

events ( outcomes ) = 6 × 6 = 36

So, the total number of outcomes = 36

- List all combinations that will give a sum of 8 below:

Number of events = {(2 , 6 ), (3 , 5 ), ( 4 , 4 ), ( 5 , 3 ) ,( 6 , 2) }

Therefore the number of favourable outcomes = 5......................(a)

- List all the combinations that will give a sum of 13

Number of events = 0

Therefore the number of favourable outcomes = 0.......................(b)

- List all the combinations that will give a sum of two numbers appearing on The top of the dice is less than or equal to 12

Number of events = { (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) }

Therefore, the number of favourable outcomes = 36..................(c)

Step 2: Calculate the probability of individual events

(i) The probability that the sum of two numbers appearing on the top of the dice is 8 = \frac{5}{36} [ \because Eq(a)]

(ii) The probability that the sum of two numbers appearing on the top of the dice is 13 = \frac{0}{36} = 0 [ \because Eq(b)]

(iii) The probability that the sum of two numbers appearing on the top of the dice is less than or equal to 12 = \frac{36}{36} = 1 [ \because Eq(c)