Math Game: Improve Ratio 9.66/10 by Guessing Better

[dot_binary]

I have a small mathematical game. A random number with uniform distribution between 1 and 10 is drawn, the player must guess any number except the the one drawn.The method I have found is by selecting the same number for every draw, the player will win 9 times out of 10. My question is , is there a better method of playing,lets say the player wins 29 times out of 30, that is a ratio of 9.66 out of 10?

 

[jbriggs444]

Can you come up with a way of playing that is worse? Can you lose this game more than one time out of ten on average even if you try?

The problem could be made somewhat more interesting if the number were not randomly drawn but was instead selected by a fiendishly clever opponent who is trying to out-guess you. A strategy of selecting the same number every time would be defeated soundly by such an opponent. Is there a strategy that you could employ in order to win 90% of the time on average against any such opponent?