Problem:
When rolling a certain unfair six-sided die with faces numbered , and , the probability of obtaining face is greater than , the probability of obtaining the face opposite face is less than , the probability of obtaining each of the other faces is , and the sum of the numbers on each pair of opposite faces is . When two such dice are rolled, the probability of obtaining a sum of is . Given that the probability of obtaining face is , where and are relatively prime positive integers, find .
Solution:
Let denote the probability of obtaining on the first die and on the second. Then the probability of obtaining a sum of is
Let the probability of obtaining face be . Then the probability of obtaining the face opposite face is . Therefore
Then , and so . The probability of obtaining face is therefore , and .
The problems on this page are the property of the MAA's American Mathematics Competitions