2024 SSMO Speed Round Problems/Problem 8
Problem
Bob has two coins; one is fair, and one lands on heads with a probability of 
Bob chooses a random coin and flips it twice. Alice watches the two coin flips and guesses whether Bob flipped the fair or rigged coin. Given that Alice is a good mathematician and guesses the more likely option (guessing randomly when they are equally likely), the probability she guesses right can be expressed as 
for relatively prime positive integers \(m\) and \(n.\) Find 
