WHAT ARE THE ODDS??

[BruceZ]

You actually don't need to know anything about poker to answer this; it is figured out logically. If all players were of equal ability, a player's chance of winning a tournament would be equal to the ratio of his chips to all the chips in the tournament. This statement does not apply perfectly to no-limit tournaments because players with short stacks have the advantage of going all-in more often, and there are effects of position. This makes the player's slightly unequal, even though we said they are of equal ability. We'll ignore that detail. To prove that players must win in proportion to their stack size, we notice that if they play a large number of freezeouts, each player must win the same number of chips in the long run since they are of equal ability. For example, in your tournament, each player initially has a 1/600 chance of winning when everyone has 10K. If I have 20K, then when I lose I lose 20K, but when I win I win 6 million. Those are the only 2 possibilities, and the details of how I reach one of those outcomes are irrelevant. I have to win 20K/6M = 1/300 of the time in order to win the same number of chips as everyone else in the long run, so my chances have doubled. Sklansky presents this proof in TPFAP. Now say my ability is greater than the average player so that I initially have a 1/100 chance of winning. Now when I get to 20K, I will again either lose 20K or win 6 million, but now I must win 6 times more chips in the long run than if all the player's were equal, so I must win 6*20K/6M = 1/50 of the time.