AIME II 2026 (Problem 7)
A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any coins can be written as , where and are relatively prime positive integers. Find .
Let be the probability that Alice and Bob each receive at least two coins before Carol receives any coins.
Let be the probability that Alice and Bob receive at least two coins and one coin respectively before Carol receives any coins.
Let be the probability that Alice and Bob each receive at least one coin before Carol receives any coins.
Let be the probability that Alice receives at least two coins before Carol receives any coins.
Let be the probability that Alice receives at least one coin before Carol receives any coins.
Define the remaining probabilities similarly.
Let be the probability that Alice and Bob receive at least two coins and one coin respectively before Carol receives any coins.
Let be the probability that Alice and Bob each receive at least one coin before Carol receives any coins.
Let be the probability that Alice receives at least two coins before Carol receives any coins.
Let be the probability that Alice receives at least one coin before Carol receives any coins.
Define the remaining probabilities similarly.
From Hint 6, so
From Hint 5,
Therefore,
Therefore,
From Hint 4,
From Hint 3,
Similarly,
Similarly,
From Hint 2,
Thus,
so
Thus,
so
754