Krishna
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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)