97 NBA draft lottery. The National Basketball Association

(NBA) utilizes a lottery to determine the order in which teams draft amateur (high school and college) players.

Teams with the poorest won–lost records have the best chance of obtaining the first selection (and, presumably, best player) in the draft. Under the current lottery system, 14 table tennis balls numbered 1 through 14 are placed in a drum, and 4 balls are drawn. There are 1,001 possible combinations when 4 balls are drawn out of 14 without regard to their order of selection. Prior to the drawing, 1,000 combinations are assigned to the 13 teams with the worst records—the lottery teams—based on their order of finish during the regular season. (The team with the worst record is assigned 250 of these combinations, the team with the second-worst record is assigned 200 of the combinations, the team with the third-worst record is assigned 157 of the combinations, etc. The table on page 150 gives the number of combinations assigned to each lottery team.) Once the 4 balls are drawn to the top of the drum to determine a four-digit combination, the team that has been assigned that combination will receive the first pick. The 4 balls are then placed back in the drum, and the process is repeated to determine the second, third, and subsequent draft picks. ( Note : If the one unassigned combination is drawn, the balls are drawn to the top again.)

a. Demonstrate that there are 1,001 four-number combinations of the numbers 1 through 14.

b. For each of the 13 Lottery teams, find the probability that the team obtains the first pick in the draft.

c. Given that the team with the second-worst record obtains the number-1 pick, find the probability that the team with the worst record obtains the second pick of the draft.

d. Given that the team with the third-worst record obtains the number-1 pick, find the probability that the team with the worst record obtains the second pick of the draft.

e. Find the probability that the team with the worst record obtains the second pick of the draft, assuming that it did not obtain the first pick.

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NBA Lottery Team Number of Combinations Worst record 250 2nd-worst record 200 3rd-worst record 157 4th-worst record 120 5th-worst record 89 6th-worst record 64 7th-worst record 44 8th-worst record 29 9th-worst record 18 10th-worst record 11 11th-worst record 7 12th-worst record 6 13th-worst record 5 Based on McCann, M. A. "Illegal defense: The irrational economics of banning high school players from the NBA draft." Virginia Sports and Entertainment Law Journal, Vol. 3, No. 2, Spring 2004, (Table 5).