There are two kinds of flash player. The flash player plug-in for your browser, and the one for your computer... Even though Adobe doesn't separate them, or even change the name they are different. If your using Firefox you it will just download the plug-in. So use a different browser. Also some games give you a link in the game to the adobe site for the update. Or just go to Google and type Flash player 10 update or something like that.
