Mock AIME I 2015 Problems/Problem 4
At the AoPS Carnival, there is a "Weighted Dice" game show. This game features two identical looking weighted 6 sided dice. For each integer , Die A has probability of rolling the number , while Die B has a probability of rolling . During one session, the host randomly chooses a die, rolls it twice, and announces that the sum of the numbers on the two rolls is . Let be the probability that the die chosen was Die A. When is written as a fraction in lowest terms, find the sum of the numerator and denominator.
The probability that A lands at 10 is The probability that B lands at 10 is The probability that A was picked is thus . The sum of numerator and denominator is thus .