I was trying to devise an equitable MLB draft lottery system which reduces the benefit of "tanking" but heavily weighs record and has a slight adjustment for lower gross revenue teams.

The "Rules":
  • 10 of 30 MLB teams make the playoffs each year.
  • All non-playoff teams receive the number of "balls" corresponding to their position--the worst record in MLB gets 30 entries, second worst gets 29, etc, up to the highest team not to win a Wild Card spot, which gets 11.
  • A second set of lottery entries based on 5-team tiers of gross revenue--the 5 highest revenue teams get 0 entries, teams 6-10 get 1, 11-15 get 2, 16-20 get 3, 21-25 get 4, and the bottom 5 revenue teams get 5 lottery balls.
  • In total, there would be 485 lottery balls. (410 for standings, 75 for revenue based).
  • The first 5 picks of the draft are randomly drawn. After that, it goes in straight record-based order. This means the lowest that the worst finishing team could pick is at #6.

A playoff team that is also a top 5 revenue team has 0% chance to move up from its position. If a low revenue team like Tampa Bay made the playoffs, it would have a 1.03% chance at the top pick. That one seemed easy, with the team having 5 balls out of 485.

However, I am unsure how to calculate the probability of such a team with 5 balls in the lottery receiving any of the lottery picks (top 5), since the number of balls removed after the first selection is variable.
  • To calculate the odds of getting a successive lottery pick, what is the correct method to calculate the average number of balls removed after the 1st pick? After the 2nd pick? Etc.
  • Is there a more elegant formula to represent this? Could it also account for the general case (a team with x number of balls)?