I studied and learned about this thing called a "Probability tree". Pretty simple, actually- a map of all possible solutions, with all the paths addiing up to 1.00. This scenario is WITHOUT replacement (otherwise wouldn't make much sense to do the draw in the first place).
My calculation is the first one to run to the front of the room and draw (note, I did that at the Don Lowe Masters back in 2004) had 1/10 chance of drawing the "1" out of the hat, and 9/10 chance it is not drawn. Second draw (next "Y" of the probablity tree), 1/9 chance of drawing the "1" multiplied by 9/10 (the probability that it was not drawn first) which comes out to 1/10 again, for the chance of drawing "1". While the probability of NOT drawing the second round is 8/9 times 9/10, which is 4 out of 5 it was not drawn. Now from a pilot persective, I already drew first, and did not get "1" so the next person that draws has twice the chances of drawing the "1" (1/10 plus 1/10) equals 2/10 or 1 out of five chances of drawing the one (note, it is this way because these are not independent events). The premise is "if-then", like Kevin said, its assuming the number 1 has not been drawn. Its a bit hard to imagine how the first pilot up has a one in 10 chance of drawing the 1, while the second pilot up has a 1 in 5 chance, the third pilot as a little better than 1 in 3 chance of drawing it! So it goes until the "1 is drawn".
Now, I noticed at many contests since, there are actually two drawings which equalize the probabilties quite a bit (so pilots like me don't game the draw by running up to be the first to draw his flight order number). A hat with all the 10 pilot's names are then mixed up, Dale pulls names out of the hat, then that pilot goes up to draw his number out of another hat for his place in the flight order.
Any way, I got fairly deep into several probability and statistics books thus the delay in response. If any of the above is not correct, please speak up. I'll continue studying the subject and will report back if I discover otherwise.