From: gidi@Hilbert.Stanford.EDU (Gidi Avrahami)
Subject: Re: Lottery
Organization: Brotherhood Of Breath
Date: Thu, 9 Apr 1992 21:05:04 GMT


[sorry if you see this more than once, I cancelled the previous version]

sdo@claude.mitre.org (Sean D. O'Neil) writes:
>Wait a minute, I'm terribly confused.  I thought that the team with the
>BEST record in the lottery got 11 balls

You were desperately wrong.  The idea is that the lottery is weighted
towards the BAD teams, not the good.  The worse you are, the better
your chances to win the lottery and draft first.

It also means that teams who are close to the playoff but just barely
miss it have a minute chance of winning the lottery.  It takes away some
of the incentive to tank games in the last month.

As for your confusion about how the T-Wolves are guaranteed to pick #4
-- the lottery has only three steps, after which the remaining teams
pick in the order of their records.  Thus, if the team with the worst
record (in our case, Minnesota) did not come up with one of the first
three, it will be the first of the rest, and pick fourth.

Confused?  Don't be.  Lets' look at last year: The eleven lottery
teams were, in order of badness:

   1   Denver        20-62
   2   Miami         24-58
   3   Sacramento    25-57
   4   Charlotte     26-56
   5   New_Jersey    26-56
   6   Dallas        28-54
   7   Minnesota     29-53
   8   Washington    30-52
   9   LA_Clippers   31-51
  10   Orlando       31-51
  11   Cleveland     33-49

I'm not sure about the tie breakers, but let's say that's how they
were ordered.  Denver had 11 balls, Miami 10, Sacto 9, Charlotte 8,
etc.., until Cleveland had only 1.  Charlotte got lucky (an event with
probability 8/66 = 0.121) and won the #1 pick.  New Jersey won the
second pick (probability 7/(66-8) = 7/58 = 0.121) and Sacramento won
the third (probability 9/(66-8-7) = 9/51 = 0.176)

Then the rest of the teams were ordered as they were, with Denver & Miami
taking the 4th and 5th spots:

   1   ^ Charlotte     26-56
   2   ^ New_Jersey    26-56
   3   ^ Sacramento    25-57
   4   * Denver        20-62
   5   * Miami         24-58
   6   Dallas        28-54
   7   Minnesota     29-53
   8   Washington    30-52
   9   LA_Clippers   31-51
  10   Orlando       31-51
  11   Cleveland     33-49

Two years ago, Seattle missed the playoffs on the last play in the
last game of the last day.  They were ranked 10th, with 2 ping pong balls.
One of these balls came out in the second round, and they picked #2.
As you can see below, this event has a 0.033 probability.


      # of     +------- Odds of picking -------------------------+
      Balls     FIRST       SECOND       THIRD      LATER   (RANGE)
  1)   11      0.16666     0.15561      0.14345    0.53428  (4)
  2)   10      0.15151     0.14471      0.13676    0.56702  (4)
  3)    9      0.13636     0.13306      0.12884    0.60174  (4)
  4)    8      0.12121     0.12070      0.11964    0.63845  (4-5)
  5)    7      0.10606     0.10766      0.10914    0.67714  (5-7)
  6)    6      0.09090     0.09396      0.09732    0.71782  (6-9)
  7)    5      0.07575     0.07967      0.08420    0.76038  (7-10)
  8)    4      0.06060     0.06479      0.06980    0.80481  (8-11)
  9)    3      0.04545     0.04936      0.05415    0.85104  (9-11)
 10)    2      0.03030     0.03340      0.03728    0.89902  (10-11)
 11)    1      0.01515     0.01694      0.01921    0.94870  (11)

The last column (in parenthesis) is like this: Say you are the 7th
team, and have 5 balls in the urn.  If the 3 lucky teams are all above
you, you remain in 7th place.  For every team *below* you that gets
lucky, you fall one step down.  In the unlikely event that all 3 lucky
teams are below you, you get pushed 3 steps down, in this case, down
to #10.


--Gidi