Calculating Odds Of Coming In Places Other Than First

I'm new to these forums, so if this has already been posted, feel free to flame and/or delete at will. But, I saw some people using approximations and stuff, and i'm 95% sure this is actually correct, so enjoy.

I think it's easiest with an example, so here's a theoretical tourny with easy numbers:

Player 1: t100
Player 2: t200
Hero: t700
Total: t1000

Your odds of first are obviously 700/1000 = 70%

Your odds of second are slightly more complicated. You get them by figuring each other player's odds of winning multiplied by the "tournament" you play out with the remaining field, then summing. Thus:

Odds of Player 1 winning and you coming in second =
(Player 1 chips) / (Total Chips) * (Hero Chips)/ (Total Chips - Player 1 chips) = 10% * 78% = 7.8%

Odds of Player 2 winning and you coming in second =
(Player 2 chips) / (Total Chips) * (Hero Chips)/ (Total Chips - Player 2 chips) = 20% * 88% = 17.6%

Total Odds of Second Place = 17.6% + 7.8% = 25.4%

Now, it follows that the odds of coming in 3rd are 4.6%. You can check for yourself that the formula is consistent with the odds of player 1 and player 2 placing in second and third. Note that for more than 3 players, the odds of coming in third can be found by the very long calculation of

SUM over X( SUM over Y (Player X) * (Player Y) * (Hero)/((Total Chips)*(Total Chips-Player X) * (Total Chips - Player X - Player Y)) ) ).


I noticed people in this forum using approximations that resulted in the following:

Player 1: 500
Player 2: 500
Hero: 9000

Hero's odds of first: 90%
Hero's odds of second: 5%
Hero's odds of third: 5%

This is clearly wrong because hero, with his monstrous (and exaggerated) chip advantage clearly has a much larger chance of coming in second than last. The real values are

1st: 90%
2nd: ~9.4%
3rd: ~.6%

Does anyone know what SnG Power Tools uses?