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#1 2007-10-05 09:32:35

agclifto
Member
Registered: 2007-10-05
Posts: 2

probability

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 the players is extremely important (see first picks in 1985 and 1992 mentioned below; see NBA Lottery Picks for a list of all players selected in the NBA draft lottery through 2006 since the system was instituted in 1985; see 2007 NBA Draft for the players selected in the 2007 draft). The order of selection is determined by a lottery drawing conducted approximately four weeks prior to the June draft; see 2007 NBA Draft Lottery News and 2007 NBA Draft Lottery Results to read about the 2007 NBA Draft Lottery held May 22, 2007. Click here to read the 2007 lottery losers' complaints that "probability" is not fair. 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 1985 a 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.

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, in 1990 the 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 1992 the 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 the 1993 draft 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?
(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 the 1994 draft, 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.)


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 the 2008 lottery the NBA assigns the 4-number configurations so that the team with the worst record had 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 2008 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).

__________

Offline

#2 2007-10-07 07:47:44

HallsofIvy
Guest

Re: probability

agclifto wrote:

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 the players is extremely important (see first picks in 1985 and 1992 mentioned below; see NBA Lottery Picks for a list of all players selected in the NBA draft lottery through 2006 since the system was instituted in 1985; see 2007 NBA Draft for the players selected in the 2007 draft). The order of selection is determined by a lottery drawing conducted approximately four weeks prior to the June draft; see 2007 NBA Draft Lottery News and 2007 NBA Draft Lottery Results to read about the 2007 NBA Draft Lottery held May 22, 2007. Click here to read the 2007 lottery losers' complaints that "probability" is not fair. 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 1985 a 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.

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).

You don't seem to be trying very hard!  There were 7 teams in the lottery and each had an equal chance: 1/7.  (If you really want 3 decimal places, you have to do the division yourself!)

________   

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, in 1990 the 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 1992 the 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 the 1993 draft 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?
(Use 3 decimal places in your answer).

Again, it is a fraction: 1 is the numerator since the Magic were given w10= 10.  The denominator is now the sum of all the weights.

_________

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).

Same thing!  A fraction with the sum of the weights as denominator.  Now what will the numerator be?

_________

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 the 1994 draft, 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).

Do you see that there are 14 different possibilities for the first ball?  Now there are 13 balls left and so 13 different possibilities for the second ball.  Similarly, 12 possibilities for the third and 11 for the last.  "Fundamental law of counting" says you multiply those together to find how many different ways of choosing 4 balls there are.  But that includes the same 4 balls in different orders.  Since you don't want to count that, divide by 4!= 4(3)(2)(1)= 24.

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