WHAT ARE THE ODDS??

[Leonardo]

I am Phil Hellmuth (not really, but lets pretend). I sit down at this years world series main event. There are exactly 600 entries, for a total of $6mill in chips on the tables. My odds of winning the event are 1 in 100 (for arguments sake). First hand. I pick up AA, move in and get called. WOW . Double up exactly to 20k. Now what are my chances of winning. Im fairly sure its in TPFAP , where Sklansky states that ones chances of winning are proportional to the chips he has (assuming equal skill levels), hence , you double your chips, you double your chances. Now has Phils odds of winning just gone up to 1 in 50?? I cant see how. It is argued that to win, you need to double up x times, and hence if you double up once already, you are twice as likely to win. But lets look at it. Before you doubled, you needed to win the remaining $5,990,000 left on the table. You figure you are 1 in 100. Now you double through, and need to win the remaining $5,980,000. Your chances have doubled?? I doubt it. Very Much. We arent flipping coins here. You dont just double or bust. You take small picks at stacks. Now, Phil with 20k, is still going to win lots of small pots on his way to 30k. He would have won the same pots with 10k, and be up to 20k. Now his stack is only 3/2 times as big. Get my drift?? Keep in mind, Sklansky knows a hell of a lot more about poker than I do, so I could very well be wrong. But my intuition tells me Im right. All comments welcome.
Leonardo

[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.