Grid Intersection

Fix a positive integer $n$ . Alice and Bob start at $(0,0)$ and $(n,n)$ , respectively. A fence has a square shape with vertices at $(0,0), (0,n), (n,0),$ and $(n,n)$ . Alice moves up or right $1$ unit with equal probability at each step, while Bob moves down or left $1$ unit with equal probability at each step. In the event that Alice or Bob is on the boundary of the fence, they take the only legal move that keeps them inside the fence. Find the probability that Alice and Bob meet before reaching the opposite corner of the fence when $n = 5$ .

Answer format: round to 3 decimal places.