Token on the Tiles

A carnival game uses a board of $8 \times 8$ painted squares, each square measuring $2$ inches per side. A player flicks a circular token of diameter $1$ inch so that its center lands uniformly at random somewhere on the board. To win, the token must sit entirely inside one painted square — touching or crossing any painted gridline, or hanging off the board's outer edge, loses. What is the player's probability of winning?