A Probability-Guided History of the National Basketball Association Player Draft.

In June the National Basketball Association (NBA) conducts its annual player draft. Each team drafts (selects) a player not yet in the NBA to be a player on their team. The order in which the teams select players is determined by a lottery. Thedraft lotteryto determine the order in which teams selected players is held in late May and televised with much drama and fanfare on ESPN. Theplayer draftis held about a month later in late June.

The NBA draft typically generates a great deal of interest in the Triangle area since players from Duke, UNC, and NC State are often candidates to be high draft picks. In the draft held in June 2009, Tywon Lawson and Tyler Hansbrough from UNC and Gerald Henderson from Duke were early round picks. In the June 2010 draft John Wall from Raleigh and the U. of Kentucky was the first player selected; in the June 2011 draft Kyrie Irving from Duke was the first player selected.

And the money isn't too bad; the first player picked in the draft is guaranteed a salary well over $4 million (as the first player selected in the 2011 draft Kyrie Irving was guaranteed a first-year salary of $4,420,900), the second player picked is guaranteed a salary of about $4 million, and so on; the 30th player selected is guaranteed a salary of about $900,000.

The order in which the teams select the players isextremelyimportant since teams choosing early can select the best players (see first picks in 1985 and 1992 mentioned below); seeNBA Draft First Picksfor a year-by-year list of the first players selected in the NBA draft. See2014 NBA Draft Lottery Resultsto see the order in which teams selected players and the players selected in the June 26, 2014 draft as determined by the May 20, 2014 draft lottery. See2015 draft lottery resultsfor the outcome of the May 20, 2015 lottery to determine the draft order in June.

Click hereto read accusations by NBA team executives that the draft lottery is "rigged" by the commissioner.

To maintain competitive balance among the teams, the NBA allows the weaker teams to select players first, in the hope that the weaker teams get the best new players.Over the years a variety of probability-based techniques have been used by the NBA to determine the draft order.

Prior to 1985, the last-place finisher in the Western Conference and the last-place finisher in the Eastern Conference would flip a coin to determine which team selected first and which team selected second.In 1985a lottery system was started to prevent the teams with the worst records from automatically receiving the first two picks; this was to prevent teams from intentionally losing games to gain a top draft pick.In this lottery system, each of the seven teams that failed to make the post season playoffs (the seven worst teams) had an equal chance of drafting first. The first year was a memorable one as the New York Knicks won the first pick and selected 7-foot center Patrick Ewing from Georgetown University. Instant success: Ewing led the Knicks to the playoffs 13 times in his 15-year career.Note: conspiracy theorists claim that the 1985 draft was rigged so that Ewing could play in New York; seethis videoand the commentary below it.

Question 1. What was the probability that the New York Knicks would win the first pick in the 1985 draft?

(Use 3 decimal places in your answer).

After a few seasons, critics pointed out that the first selection in the draft very seldom had been awarded to the worst or second-worst team in the league. As a result, in1990the NBA changed the draft lottery to a weighted probability system. By 1990 there were 28 teams in the NBA and 17 teams made the playoffs. The eleven worst teams that did not make the playoffs would be given the opportunity to choose early in the draft. This was accomplished by the assignment of weights to the eleven non-playoff teams. The team with the worst record during the regular season received a weight of w₁₁= 11, the second-worst team received a weight of w₁₀= 10, the third-worst team received a weight of w₉= 9, and so on; the team with the best record among the 11 non-playoff clubs received a weight of w₁= 1.

In 1992the Orlando Magic had the second-worst record (21-61) in the league and thus were assigned the weight w₁₀= 10. They won the first pick in the draft lottery and selected monster 7'1" center Shaquille O'Neal from LSU. With "Shaq" on the team, Orlando improved to 41-41 the next season and just missed the playoffs. So in the1993draft lottery, Orlando was assigned the weight w₁= 1 since they were the best of the eleven non-playoff teams.The Orlando Magic again won the first pick!They selected forward Chris Webber from the University of Michigan.

Question 2. What was the probability that the Orlando Magic would win the first pick in the 1992 draft lottery? (Hint: view this as randomly selecting a slip of paper from a hat; the worst team has its name on 11 slips of paper in the hat, the second-worst team has its name on 10 slips of paper in the hat, and so on; the best of the eleven non-playoff teams has its name on 1 slip of paper. How many slips of paper are in the hat?)

(Use 3 decimal places in your answer).

Question 3. What was the probability that the Orlando Magic would win the first pick in the 1993 draft lottery?

(Use 3 decimal places in your answer).

The probability-challenged NBA executives did not fully comprehend the rarity of the occurrence of Orlando winning the first pick in the 1993 draft, so the system was changed yet again. For the1994draft, 14 balls numbered 1 through 14 were placed in a drum, and 4 were chosen without replacement; the order in which the balls were drawn made no difference.

Question 4. How many ways can 4 balls be selected from 14 numbered balls if the balls are selected without replacement and order makes no difference?

(Do not use a comma to separate the digits in your answer).

To determine the draft order in all player drafts since 1994, one of the possible configurations of 4 numbered balls in question 4 is discarded and the remaining configurations are allocated to the non-playoff teams based on their order of finish during the regular season (in the 28-team NBA in 1994 and 1995 there were 11 non-playoff teams; in the 30-team NBA from 1996 to 2003 there were 13 non-playoff teams; since 2004 when the Charlotte Bobcats were added, there have been 14 non-playoff teams).

• The draft order is then determined as follows:
• 4 balls are randomly selected without replacement from the 14 numbered balls; the team that has been assigned those 4 numbers receives the number one pick
• the 4-number configurations that were assigned to the team awarded the number one pick are excluded from further consideration
• the 4 balls that determined the number one pick are placed back in the drum; 4 balls are randomly selected without replacement from the 14 numbered balls until a non-excluded 4-number configuration determines which team is awarded the number two pick
• the 4-number configurations that were assigned to the team awarded the number two pick are excluded from further consideration
• the 4 balls that determined the number two pick are placed back in the drum; 4 balls are randomly selected without replacement from the 14 numbered balls until a non-excluded 4-number configuration determines which team is awarded the number three pick

(Note: If the one discarded 4-number configuration that is not assigned to any team is drawn, the 4 balls are placed back in the drum and another selection of 4 balls is made.)

As an example,here(SCROLL DOWN) are the 199 4-number combinations assigned to the Boston Celtics in the 2007 draft.

For a more detailed explanation of the current version of the player draftsee here

Question 5. If the NBA wants the non-playoff team with the best record to have a .005 probability of winning the first pick in the draft, how many 4-number configurations should be assigned to this team? (remember that one of the possible 4-number configurations from question 4 is not used)

Suppose that in the2014lottery the NBA assigns the 4-number configurations so that the team with the worst record has a .25 probability of winning the first draft pick, the team with the second-worst record has a .20 probability of winning the first draft pick, and the team with the third-worst record has a .157 probability of winning the first pick.

Question 6. Suppose the first 4-number configuration drawn in the 2014 draft lottery is a configuration that has been assigned to the team with the third-worst record; therefore the team with the third-worst record would be awarded the first pick in the draft. What is the probability that the team with the worst record is awarded the second pick in the draft?

(Use 3 decimal places in your answer).