The NBA lottery randomly draws four numbers from the set {1, 2, 3, ..., 14} to chose its first, second, and third place draft pick. The number of chances is the number of outcomes assigned to each team. Verify the probabilities for Pick 1 and Pick 2, if each drawing which does not lead to a new outcome is redrawn. The probabilities below are for when there were 13 teams that did not make the playoffs. See http://en.wikipedia.org/wiki/NBA_Lottery for the current probabilities, now that there are 14 teams that do not make the playoffs.

The probabilities for getting the first pick are easy. They are found by using conditional probability. Let x_i be the number of outcomes assigned to team i.
Then x_i /1001 is the probability that the first drawing of four balls gives team i the first pick in the draft. But another way team i can get the first pick in the draft is if the first draw is the set of four numbers not assigned to any team, leading to a redraw, and then the redraw is one of the outcomes assigned to team i. This has probability (1/1001)×(x_i /1001).

We must in theory consider every possible way that team i can get the first pick. The first two drawings may both be the set of four numbers not assigned to any team, then the third draw is for team i. This has probability (1/1001)×(1/1001)×(x_i /1001).
And so on. These probabilities must be summed to get the probability that team i gets the first pick. This gives an infinite geometric series, which can be summed by an elementary formula:
x_i /1001 + (1/1001)×(x_i /1001) + (1/1001)×(1/1001)×(x_i /1001) + (1/1001)×(1/1001)×(1/1001)×(x_i /1001) + ...
= x_i /1001×[1/(1-1/1001)] = x_i /1000
which was what we should have expected since only 1000 outcomes have been assigned, and each outcome is equally likely. So for example, the team with 250 outcomes assigned to it has a 250/1000 = 25% chance of getting the first pick in the draft.

To see the work for finding the probabilities of getting the second pick in the draft,
click on page 2.