Finding the probabilities of the getting the first pick was a warmup to finding the probability of getting the second pick in the draft.
Again let's focus on finding the probability team i gets the second pick. One way is if team j gets the first pick (with probability xj /1000, from above), and then the following draw is one of the outcomes assigned to team i (with probability xi /1001). This outcome has probability (xj /1000)×xi /1001).
Or if the second draw (first draw after picking the team to get the first draft pick) is one of the numbers assigned to team j (the team that got the first pick), or if the second draw is the one number not assigned to any team, then there will be a redraw. This has probability (xj +1)/1001. So the probability the first pick goes to team j, then the next draw leads to a redraw, then the draw after that goes to team i is (xj /1000)×(xj +1)/1001×(xi /1001).
The probability the first pick goes to to team j, then the next two draws leads to a redraw, then the draw after that goes to team i is (xj /1000)×(xj +1)/1001×(xj +1)/1001×(xi /1001).
Again, to find the probability team j gets the first pick and team i gets the second pick in the draft, we get an infinite geometric series:
(xj /1000)×xi /1001) + (xj /1000)×(xj +1)/1001×(xi /1001) + (xj /1000)×(xj +1)/1001×(xj +1)/1001×(xi /1001) + ...
= [(xi ×xj )/(1000×1001)]×[1 + (xj +1)/1001 + (xj +1)2/10012 +...]
= [xi /1000]×[xj /(1000 - xj )]
Now to compute the probability that team i gets the second pick, we must sum over all the teams j not equal to i.
So the team with 250 outcomes assigned to it has
250/1000×[200/800 + 157/843 + 120/880 + 89/911 + 64/ 936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] =
4191481375219641521698767977/19447133696754332942062465776 ~= .2155320903, or about a 21.55% chance of getting the second pick in the draft.
The team with 200 outcomes assigned to it has
200/1000×[250/750 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] =
2298314996951011728995868007/12154458560471458088789041110 ~= .1890923389, or about a 18.91% chance of getting the second pick in the draft.
The team with 157 outcomes assigned to it has
157/1000×[250/750 + 200/800 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] =
2741420664183576337125803293/17301720370777876282973724000 ~= .1584478656, or about a 15.84% chance of getting the second pick in the draft.
The team with 120 outcomes assigned to it has
120/1000×[250/750 + 200/800 + 157/843 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] =
468100489364942042002509229/3683169260748926693572436700 ~= .127091767, or about a 12.71% chance of getting the second pick in the draft.
The team with 89 outcomes assigned to it has
89/1000×[250/750 + 200/800 + 157/843 + 120/880 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] =
521407593415555127753226431/5336754581985272486844804000 ~= .09770124996, or about a 9.77% chance of getting the second pick in the draft.
The team with 64 outcomes assigned to it has
64/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] = 281007498159344351672031542/3895659795022903233586231125 ~= .07213348006, or about a 7.21% chance of getting the second pick in the draft.
The team with 44 outcomes assigned to it has
44/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] = 23382057570276729034705471/462322501349237660281059000 ~= .05057520995, or about a 5.06% chance of getting the second pick in the draft.
The team with 29 outcomes assigned to it has
29/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 18/982 + 11/989 + 7/993 + 6/994 + 5/995] = 169247501963720350392846431/5006986018731805597853364000 ~= .03380227173, or about a 3.38% chance of getting the second pick in the draft.
The team with 18 outcomes assigned to it has
18/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 11/989 + 7/993 + 6/994 + 5/995] = 11655722315351872859801659/550099957477775880913738000 ~= .02118837160, or about a 2.12% chance of getting the second pick in the draft.
The team with 11 outcomes assigned to it has
11/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 7/993 + 6/994 + 5/995] = 5822043896341714357778221/446896169150527000231236000 ~= .01302773283, or about a 1.30% chance of getting the second pick in the draft.
The team with 7 outcomes assigned to it has
7/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 6/994 + 5/995] = 17455599574758064759521899/2098309634954071314421932000 ~= .008318886443, or about a 0.83% chance of getting the second pick in the draft.
The team with 6 outcomes assigned to it has
6/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 5/995] = 11635269755969099813885977/1630376735140369964961642000 ~= .007136552862, or about a 0.71% chance of getting the second pick in the draft.
The team with 5 outcomes assigned to it has
5/1000×[250/750 + 200/800 + 157/843 + 120/880 + 89/911 + 64/936 + 44/956 + 29/971 + 18/982 + 11/989 + 7/993 + 6/994] = 29083642097818138452860831/4886214496671942950266951200 ~= .005952182844, or about a 0.60% chance of getting the second pick in the draft.
To find the probability of getting the third pick in the draft would involve summing over all possible ways of team j picking first, team k picking second, then team i getting the third pick (j not equal to i, k not equal to j or i). I'll do that when I'm paid for that task :o)
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