Project #14109 - statistics

  • Description

    Homework #6 deals with basic topics in counting, random variables, expected value, standard deviation, and binomial random variables.This material is covered in chapters 15, 16, and 17 in the text.

  • Instructions

    Homework #6

    You are allowed four submissions.

    Do not wait until the last minute to start this homework!

 

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

 

 

 
Question Part
Points
Submissions Used
1 2 3 4 5
/1 /2 /2 /1 /2
0/4 0/4 0/4 0/4 0/4
Total
/8
 

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. The lottery drawing is conducted approximately four weeks prior to the June player draft. The 2011 draft lottery to determine the order in which teams selected players was held May 17 and televised with much drama and fanfare on ESPN. The 2011 player draft was held June 23.

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 2011 draft (Kyrie Irving) is guaranteed a first-year salary of $4,420,900, the second player picked is guaranteed a salary of $3,955,400, and so on; the 30th player selected is guaranteed a salary of $877,300.

The order in which the teams select the players is extremely important since teams choosing early can select the best players (see first picks in 1985 and 1992 mentioned below); see NBA Draft First Picks for a year-by-year list of the first players selected in the NBA draft. Click 2012 NBA Draft Lottery Results to see the order in which teams selected players in a recent draft as determined by the May draft lottery. See First Round Selections for a list of the players selected in the first round of the 2012 draft.

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

The current lottery-based system is the most recent of several different methods used by the NBA to determine the order in which teams select new players. Over the years a variety of probability-based techniques have been tried by the NBA to determine the draft order. Below is a brief history of the NBA draft with an emphasis on the probability issues associated with the various versions of the draft.

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? (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 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 (note: 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.)

To see the 199 4-number combinations assigned to the Boston Celtics in the 2007 draft see here.

For a more detailed explanation of the current version of the player draft see here

Question 5. If the NBA wants the non-playoff team with the worst record to have a .25 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)


 

 
Question Part
Points
Submissions Used
1 2 3 4
/1 /1 /1 /1
0/4 0/4 0/4 0/4
Total
/4
 

Define the random variable X as the number of runs scored in a half-inning by a major league team in the National League (a half-inning consists of 3 outs). Define Y as number of runs scored in a half-inning by a major league team in the American League. The probability distributions of X and Y are shown in the table below (determined from data for several recent seasons). The probability of scoring more than 5 runs in a half-inning is negligible and is not included.

Runs scored in half-inning 0 1 2 3 4 5
Probability Nat'l Lg. 0.6194 0.2967 0.0711 0.0113 0.0014 0.0001
Probability Amer. Lg. 0.6108 0.3011 0.0742 0.0122 0.0015 0.0001

Question 1. On average Y, the runs an American League team scores in a half-inning, is greater than X, the runs a National League team scores in a half-inning. Determine the expected value and standard deviation of how much Y exceeds X.

expected value of how much Y exceeds X (use 4 decimal places).
standard deviation of how much Y exceeds X (use 4 decimal places).

Question 2. In an entire game between an American League team and a National League team where each team bats nine times, by how much would you expect the American League team to win? What is the standard deviation of the winning margin of the American League team win?

expected value of winning margin of American League team (use 4 decimal places)
standard deviation of the winning margin of the American League team (use 4 decimal places).
 

 

 

 

Question Part
Points
Submissions Used
1 2
/1 /1
0/4 0/4
Total
/2
 

lebron

In one of his final seasons with the Cleveland Cavaliers Lebron James was accused by the fans of "choking" in the NBA playoffs. The primary reason for this accusation was Lebron's three-point shooting percentage in playoff games.

During the regular season Lebron had made 31.5% of his three-point shot attempts. However, in the playoffs Lebron took 70 three-point shots and made 18.

Question 1. If we assume Lebron's ability to make a three-point shot really was 31.5%, what is the probability that Lebron would make 18 or fewer three-point shots in 70 attempts?



Question 2. If we assume Lebron's ability to make a three-point shot really was 31.5%, how many three-point shots would you expect Lebron to make in 70 attempts?

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Due By (Pacific Time) 10/08/2013 12:00 am
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