probability

Question

Two six sided dice are rolled. What is the probability that the sum of the numbers rolled is a multiple of 3 or the dice sum to 9?

Answer 1

For a single die roll there are 6 possible outcomes {1,2,3,4,5,6}.  When you roll two dice there are 36 possible outcomes, 6 on the 1st die and 6 on the 2nd, which can be listed as 

 

(1,1), (2,1), (3,1), (4,1), (5,1), (6,1), 

(1,2), (2,2), (3,2), (4,2), (5,2), (6,2),

(1,3), (2,3), (3,3), (4,3), (5,3), (6,3),

(1,4), (2,4), (3,4), (4,4), (5,4), (6,4),

(1,5), (2,5), (3,5), (4,5), (5,5), (6,5),

(1,6), (2,6), (3,6), (4,6), (5,6), (6,6) 

 

There are 2 outcomes with a sum of 3, namely (1,2) and (2,1)

There are 5 outcomes with a sum of 6 {(1,5),(2,4),(3,3),(4,2),(5,1)}

There are 4 outcomes with a sum of 9 {(3,6),(4,5),(5,4),(6,3)}

There  is  1 outcome with a sum of 12, namely (6,6) 

 

These are all the outcomes with the sum being a multiple of 3 or equal to 9, so the probability that this occurs is 

(2 + 5 + 4 + 1) / 36 = 12/36 = 1/3