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.
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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 w_{11} = 11, the second-worst team received a weight of w_{10} = 10, the third-worst team received a weight of w_{9} = 9, and so on; the team with the best record among the 11 non-playoff clubs received a weight of w_{1} = 1.
In 1992 the Orlando Magic had the second-worst record (21-61) in the league and thus were assigned the weight w_{10} = 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 w_{1} = 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).
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)
Go to the website Mega Millions Lottery which gives results and information for the Mega Millions lottery. In this game the player must choose five numbers from the numbers 1 to 56 and one "Mega Ball number" from the numbers 1 to 46.
Question 1. In the Mega Millions Info table what should the number be at the end of the Jackpot row? (the number posted on the website is incorrect)
1 in
Question 2. Suppose that Mega Millions game officials change the game as follows: the player must choose six numbers from 1 to 53 and one "Mega Ball" number from the numbers 1 to 43. The odds for this new game are
1 in
Define the random variable Y as the number of runs scored by a major league team in a half-inning (a half-inning consists of 3 outs). The probability distribution function of Y is shown in the table below (determined from TeamRankings.com). 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 | 0.6126 | 0.3002 | 0.0735 | 0.0120 | 0.0015 | 0.0001 |
Question 1. What is the probability that a team will score 2 or more runs in a half-inning?
Question 2. Calculate E(Y), the expected number of runs scored by a team in a half-inning.
(use 4 decimal places)Question 3. Calculate SD(Y), the standard deviation of the number of runs scored by a team in a half-inning.
(use 4 decimal places)Question 4. Calculate the expected number of runs scored in one inning.
(use 4 decimal places)Question 5. Calculate the standard deviation of the number of runs scored in one inning.
(use 4 decimal places).In the 4x100 medley relay event in swimming, four swimmers swim 100 yards, each using a different stroke. The four medley relay swimmers for a college team preparing for their conference championships have the following summary statistics (in seconds) for the four strokes:
Swimmer | Mean | St. Dev. |
1 (backstroke) | 50.72 | 0.26 |
2 (breaststroke) | 55.51 | 0.29 |
3 (butterfly) | 49.43 | 0.2 |
4 (freestyle) | 44.91 | 0.21 |
Question. Let the random variable T denote the relay team's total time in the medley event. Determine the mean E(T) and standard deviation SD(T).
meanDefine 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).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)Recent data show that women's ice hockey is the most dangerous NCAA sport in terms of concussions, more dangerous than men's ice hockey and even football (data from NYTimes and NCAA). This result is even more surprising when one considers that checking is not allowed in women's ice hockey.
The first table below shows the probability distribution of the number of concussions per 1,000 player hours for NCAA women's ice hockey. The second table shows the probability distribution for NCAA men's ice hockey.
Number of Concussions per 1,000 player hours | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
Probability | 0.06587 | 0.17918 | 0.24368 | 0.22094 | 0.15024 | 0.08173 | 0.03705 | 0.01440 | 0.00489 | 0.00148 | 0.00040 | 0.00010 | 0.00002 |
Number of Concussions per 1,000 player hours | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Probability | 0.22993 | 0.33799 | 0.24842 | 0.12173 | 0.04473 | 0.01315 | 0.00322 | 0.00068 | 0.00012 | 0.00002 |
Buffy and her boyfriend Bubba both play college hockey at the same university.
Question. Over 1,000 player hours what is the probability that the number of concussions on Buffy's team is within 1 standard deviation of the mean? (mean is another name for the expected value)
Answer the same question for Bubba's team.
Note: calculate the means and standard deviations to 2 decimal places.
In 1998, Mark McGwire of the St. Louis Cardinals hit 70 home runs, a new major league record at the time. Was this feat as surprising as most of us thought? In the three seasons before 1998, McGwire hit a home run in 11.6% of his times at bat. He went to bat 509 times in 1998. If he continues his past performance, McGwire's home run count in 509 times at bat can be modeled as a binomial distribution with n = 509 and p = 0.116.
Question 1. What is the mean number of home runs McGwire will hit in 509 times at bat?
Question 2. Use the binomial distribution to find the probability that McGwire hits 70 or more home runs.
(Use 4 decimal places).Question 3. In 2001, Barry Bonds of the San Francisco Giants hit 73 home runs, breaking McGwire's record. This was surprising. In the three previous seasons, Bonds hit a home run in 8.65% of his times at bat. He batted 476 times in 2001. Consider his home run count as a binomial random variable with n = 476 and p = 0.0865. Use the binomial distribution to find the probability that Bonds hits 73 or more home runs.
(Use 6 decimal places).
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?
Subject | General |
Due By (Pacific Time) | 10/08/2013 12:00 am |
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