2025 SSMO Relay Round 5 Problems/Problem 2

Problem

Let Ethan places red cards and blue cards in a row so that the leftmost card is red and every second card is red. He performs a sequence of operations called replacements: in each replacement, he randomly selects a card other than the leftmost one, discards it, and moves the leftmost card to the discarded card’s position. This process is repeated until only one card remains. The probability that the final remaining card is red is for relatively prime positive integers and . Find .