Random chess position

How likely is a random arrangement of the chess pieces on the chessboard to be a legal position?

Take a standard complete set of 32 chessmen and cast them across the chessboard, so that they are scattered to random squares. Let us assume that each chessman lands on a distinct square, and let us consider the possible arrangements of the pieces on the board that can arise this way, considered uniformly at random.

Question. How likely is the resulting arrangement to be a legal position, one that could have arisen in a game of chess?