[EnderW27]

You have to be very careful how you phrase questions like this.

If the same game were played a second time the person who won the first game would have a 10% chance of winning because his first win is already a given.
Even if you asked "What are the odds that one player wins twice in a row" the answer would STILL be 10% because someone must win the first game.

To get the 1% you're looking for you must ask "starting now, what are the odds player 1 wins the next two games?"
If there were 20 cards and 20 people playing, your odds on each game is 1/20 and yes, you keep multiplying that number by itself to determine streaks. So game two you'd win 1/400 times.