A scratch game has 16 covered squares, below which are, in some order, symbols representing 9 players (a pitcher, a catcher, and so on), 3 strikes (strike 1, strike 2, and strike 3), 1 hit, 1 home run, 1 bat, and 1 ball.

You start scratching squares randomly, and if you manage to get the hit and the home run you win (never mind other symbols) unless you already got the three strikes, because then you're out!

What are the odds of winning at this game?

Computer solutions welcome!