Sarah stands at $(0,\ 0)$ and Rachel stands at $(6,\ 8)$ in the Euclidean plane. Sarah can only move $1$ unit in the positive $x$ or $y$ direction, and Rachel can only move $1$ unit in the negative $x$ or $y$ direction. Each second, Sarah and Rachel see each other, independently pick a direction to move at the same time, and move to their new position. Sarah catches Rachel if Sarah and Rachel are ever at the same point. Rachel wins if she is able to get to $(0,\ 0)$ without being caught; otherwise, Sarah wins. Given that both of them play optimally to maximize their probability of winning, what is the probability that Rachel wins?