The mistake is this:
Originally posted by davodka
All chips are tickets and put into a drum, with 1 ticket drawn for first place. With 1500 chips we have 1/9 of the tickets available, to win 50% of the prize pool.
After this draw, ALL the winner's 1500 tickets/chips are removed, leaving 8 x 1500 chips left, with 1 ticket being drawn for second place. With 1500 chips we have 1/8 of the tickets available, to win 30% of the prize pool.
In the second paragraph of the quote, you always have your 1500 tickets in the drum.
But that's not correct, because sometimes they were removed when you won the first prize.
They are in the drum only when you didn't win, which happens 8 times out of 9 (you win once out of 9). Therefore you have to count the chance of getting 2nd place as (8/9)*(1/8) = 1/9 (the 8/9 is the probability that you didn't win).
Similar thing goes for the 3rd place: you didn't win and didn't come 2nd, therefore your tickets are in the drum in (8/9)*(7/8) cases (now it's just 7/8, because we already know that you didn't win when looking at this case), therefore the total probability of finishing 3rd is (8/9)*(7/8)*(1/7) = 1/9 again.
Grand total of 1/9 * 50 % + 1/9 * 30 % + 1/9 * 20 % = 11.1 %.