Question

A pair of fair dice is rolled. Find the probability that the sum of the two numbers facing up is less than 10.

Answer 1

Let the combinations of numbers that total up sums less than 10 are : (x, y)

   x is the number facing up on the first dice.
   y is the number facing up on the second dice.

Let p = probability that  x + y < 10  = P(x+y <10)
    = 1 - P (x+y>= 10)
    = 1 - P [  (x = 4 AND y = 6 )  OR  ( x = 5 AND y = 5 )  OR (x=6 AND y = 4) ]
   = 1 -  [  P(4,6) + P(5,5) + P(6,4) ]
    =  1 -  [  1/6 * 1/6 + 1/6*1/6 + 1/6 * 1/6 ]
    = 1 - 3/36
    =  11/12  

   the probabilities are multiplied, above,  as they (results of two dice rolling)  are independent events.