2005 Alabama ARML TST Problems/Problem 1
Problem
Two six-sided dice are constructed such that each face is equally likely to show up when rolled. The numbers on the faces of one of the dice are 1, 3, 4, 5, 6, and 8. The numbers on the faces of the other die are 1, 2, 2, 3, 3, and 4. Find the probability of rolling a sum of 9 with these two dice.
Solution
We use generating functions to represent the sum of the two dice rolls:
The coefficient of 
, that is, the number of ways of rolling a sum of 9, is thus 
, out of a total of 
possible two-roll combinations, for a probability of 
.
Alternatively, just note the possible pairs which work: 
and 
are all possible combinations that give us a sum of 9 (where we count 
twice because there are two different 3s to roll). Thus the probability of one of these outcomes is 
.
