Reality Math
Joe Sulock, University of North Carolina at Asheville
Dot Sulock, University of North Carolina at Asheville
Unit 21. The NBA Draft
Major professional-team sports typically use some type of draft to help determine which teams get
the best available new professional players for the next season.
1. Reverse Order Drafts or Lottery?
Most leagues use what is called a reverse-order draft. The team with the worst record gets to choose
first, the team with the second-worst record chooses next, and so on. The NBA, though, since 1985
operates differently. The draft of amateur players is determined—at least partly—by a lottery.
Every team that does not make the playoffs is eligible for the lottery. In 1985 and 1986 there were
seven “non-playoff” teams. The order that these teams could draft was determined randomly and
each team had an equal chance of being selected. No, team, though, could win twice. For example,
in 1985 the New York Knicks got the first choice (and selected the legendary Patrick Ewing). Thus,
New York was ineligible to be selected for picks 2-7. After the first seven picks, the rest of the first
round was based on won/lost records (teams picked in reverse order of their records). And when the
first round was complete, the selection order of the remaining rounds reverted to a reverse-order
draft. Back then, the draft had a total of seven rounds.
In 1985 and 1986
1. (a) What was the probability for each one of the worst seven teams that it would get the
first pick in the draft? (b) If one of these teams was not selected to make the first pick, what is the
probability it will get the second pick? (c) If a team is not selected to pick first or second, what was
the probability it will get the third pick?
Why did the NBA start such a lottery? Essentially the NBA used a reverse-order draft prior to 1985.
Allegedly some of the weaker teams were deliberating trying to lose games in order to increase their
draft position.
This new system discouraged shirking, but it did not necessarily help out the very weakest teams.
2. If your favorite team had the worst won/lost record in the NBA, what was the worst
possible pick your team would get in the first round of the draft in 1985 and 1986?
Since 1987, the NBA has used a lottery for all the non-playoff teams to determine the first three
picks of the first round. After that, a reverse-order draft is used.
3. Your favorite team has the worst won/lost record in the NBA. What is the worst possible
pick your team will get now in the first round of the draft?
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Though only the first three picks are determined by lottery, the NBA has tweaked the probabilities
over the years. In 1987 and 1988, each non-playoff team had an equal chance of selection. In both
years, though, only one of the three weakest teams ended up with a “lottery pick.”
4. Suppose there were ten non-playoff teams and three lottery picks. Your favorite team was
one of these worst ten teams. What is the probability that it won’t be chosen to make a lottery
selection?
2. The First Lottery Scheme
Starting with the 1990 draft, the NBA abandoned its policy of giving each non-play off team an
equal chance of being selected. It decided to use what can be called a “weighted lottery system.”
The team with the worst-record would have the best probability of winning picks in the lottery, and
the non-playoff club with the best record would have the least probability of winning picks. So how
did the NBA decide these probabilities? They chose a method that, to someone, had intuitive appeal.
Back then, there were eleven non-playoff teams. The team with the worst record received 11
chances to be selected in a random process. The next-worst team got 10 chances of being selected.
The chances fell this way and the non-playoff club with the best record got only one chance. All this
means that the random process of selection will be based on a total of 66 chances.
5. Where did the number 66 come from?
The table below shows the 11 non-playoff teams for the 1990 draft along with their won/lost record
and the number of chances each team has. As before, once a team gets a first-round pick
TEAM
New Jersey
Miami
Orlando
Charlotte
Minnesota
Sacramento
L.A. Clippers
Washington
Golden State
Seattle
Atlanta
RECORD
17-65
18-64
18-64
19-63
22-60
23-59
30-52
31-51
37-45
41-41
41-41
CHANCES
11
10
9
8
7
6
5
4
3
2
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6. Find the probability that Miami will have the first pick in the draft (a) as an unreduced
fraction (b) as a reduced fraction (c) as a decimal (d) as a percent (see Appendix C)
7. Find the probability (b) that Miami will have the second pick if Seattle has the first pick.
(c) that L.A. will have the third pick if Charlotte and Orlando have the first two picks.
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8. Suppose there are 14 non-playoff teams. What is the probability that the team with the
worst won/lost record will pick first?
The NBA used the method just described in the 1990, 1991, 1992 and 1993 drafts. The NBA then
decided to tweak the probabilities again. Why? A good guess is the good luck of the Orlando
Magic. In 1992, the Magic had the second-worst record (21-61) and had a 15% probability (10
chances out of 66) of getting the top pick. In fact, they did get the first pick and selected Shaquille
O'Neal. The Magic improved next season to a record of 41-41which was the best record of any nonplayoff team. In the 1993 draft, they had only a 1.5% probability (1 chance out of 66) of getting the
top pick. But they won again! (For basketball buffs: they selected Michigan’s Chris Webber and
promptly traded him).
At that point the NBA decided that the worst teams had too few chances and the better non-playoff
teams had too many. They adopted a method that, with only a few very minor changes, is still in
effect today. Here’s how it works.
There are fourteen non-playoff teams. Fourteen ping-pong balls are numbered one to fourteen and
placed in a weird machine which randomly selects four balls. For example, maybe 11-2-7-14 are
selected.
3. Combinations
At this point, we best explain a word whose popular meaning is quite a bit different from its
mathematical meaning. In mathematics, the word “combination” means that the order or sequence
of chosen numbers doesn’t matter. Thus, if the machine selects 2, 7, 11, 14, it is equivalent to
selecting, say, 7, 2, 11, 14. What matters is that the “set of four” contains 2, 7, 11 and 14 in some
order. The order of selection is irrelevant with combinations.
Unfortunately, in popular usage, the word “combination” implies that the order of the numbers
matters and, perhaps, matters a great deal. For example, if you have the combination to a lock, you
best enter the numbers in the right sequence or the lock won’t open! Thus a combination to a lock
isn’t really a mathematical combination, it is actually a mathematical permutation with order
mattering.
A four-card hand of cards is a good example of a combination.
Jack of Diamonds, Queen of Spaces, King of Hearts, Ace of Spades = JD, QS, KH, AS
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is the same hand as QS, JD, KH, AS and the same hand as KH, AS, JD, QS and so forth. The order
of the cards in the hand doesn’t matter since the player can rearrange them at will. So having these
four cards in any order is one combination.
So, how many possible four-ball combinations of 14 ping-pong balls are there? It turns out there are
1,001 different combinations of 14 things. Remember that 1-2-3-4 is the same combination as 1-32-4, but 3-2-4-5 is a different combination. How did we get 1001? Let’s think about it.
There are 14 possible numbers on the first ball. Since the first ball is not replaced, there are 13
possible numbers on the second, 12 on the third and 11 on the fourth. If the order of the numbers
mattered there would be 14*13*12*11 = 24,024 different outcomes. Those different orderings of 4
chosen from 14 possibilities are called permutations.
If you have any trouble seeing why the multiplication above gives the answer, visualize a tree
diagram with 14 choices for the first branch, then 13 branches off each of the 14 branches giving
14*13 different results for the first two drawings. It is hard to picture the whole tree with 24,024
different branches at the top!
But, as we noted, the order doesn’t matter. We only care about the “combinations.” How to adjust
this number for the fact that 2, 7, 11, 14 is the same combination as 7, 2, 11, 14? In other words,
how to adjust for all the permutations that are the same combination? We need to divide 24,024 by
4*3*2*1 = 24.
This adjustment has some appeal if we think about it. Consider the numbers 2, 7, 11 and 14. In how
many different orders could these numbers have been drawn? Imagine you randomly picked one of
these numbers. There are four numbers you could get. If you don’t replace the number, when you
pick the second there are three possible numbers left. When you pick again, two are left. For the
fourth number, only one is left. In short, there are 4*3*2*1 = 24 different orders in which these four
numbers could have been drawn.
9. List all 24 different orders for drawing 2, 7, 11, and 14.
C414
is a common notation for the number of combinations of 4 items picked from 14 choices
C414 = 14  13 12  11
is pretty easy to get the hang of!
4  3 2  1
10. What if there were 20 non-playoff teams and four numbers were still drawn, how many
different combinations would there be?

11. What if there were 21 non-playoff teams and five numbers were drawn, how many
different combinations would there be?
12. How many different groups of three can be chosen from 20 people?
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13. How many different five-card poker hands have (a) all spades? (b) all cards from same
suit?
14. How many different five-card poker hands are there?
15. What percent of five-card poker hands are (a) all spades? (b) all the same suit?
16. What is the probability of a five-card poker hand being (a) all spades? (b) all the same
suit?
17. If there are ten men and ten women and a group of three is chosen randomly from the 20
people, what is the probability that the group contains all men? Do it by counting.
(a) what is the total number of groups of three? (b) how many groups of three contain all
men? (c) what is the probability of the group of three being all men?
4. The Amazing Lottery Scheme
Let’s return to NBA draft. How many possible four-ball combinations, as the term is used in math,
are there when you have a set of 14 different numbers?
14  13 12  11
= 1001
4  3 2  1

Now that the NBA knows there are 1001 possible combinations of four balls, what do they do next?
First, one combination is eliminated. If the machine happens to draw this combination, the process
is redone.
18. Suppose 11-12-13-14 is the eliminated combination. What is the probability that this
combination will be drawn and the process will have to be redone?
The other 1000 combinations (“chances”) are assigned to each non-playoff team based on their
won/lost record. For example, see table below.
2006 Lottery
TEAM
Portland
New York
Charlotte
Atlanta
Toronto
Minnesota
Boston
RECORD
21-61
23-59
26-56
26-56
27-55
33-49
33-49
COMBINATIONS
250
199
138
137
88
53
53
5
Houston
Golden State
Seattle
Orlando
NO/Okla. City
Philadelphia
Utah
34-48
34-48
35-57
36-46
38-44
38-44
41-41
23
22
11
8
7
6
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19. What was the chance that each would get the first pick? (a) Toronto (b) Portland
SECAUCUS, N.J., May 23 -- Following are the results from the 2006 NBA Draft
Lottery, which was conducted this evening at NBA TV’s studio in Secaucus, New Jersey.
The Toronto Raptors, who had an 8.8 percent chance of obtaining the first selection, will
have the first overall pick in the 2006 NBA Draft, which will be held in New York City
at The Theater at Madison Square Garden on Wednesday, June 28.
1. Toronto
2. New York (To Chicago)
3. Charlotte
4. Portland 5. Atlanta
6. Minnesota
7. Boston
8. Houston
9. Golden State
10. Seattle 11.
Orlando
12. New Orleans/Oklahoma City
13. Philadelphia
14. Utah
“Colangelo and his Raptors, despite having a mere 8.8 percent chance of landing the top
pick in this year’s Draft, beat the odds when the winning combination of 4—10—11—13
surfaced from the hopper.”
http://www.nba.com/features/lottery2006_index.html
20. (a) How was the 8.8% chance of the Raptors landing the top pick determined?
(b) What was the winning combination?
So Your Team Didn’t Get The Top Pick! Let’s suppose that Boston is your favorite team. And it
wasn’t selected to make the first pick. What’s the chance that Boston will win the second overall
pick? Well, it depends on who selects first since no team can win twice.
21. After Toronto won the top pick, (a) what was the probability that Boston would win the
second? (b) What team did win the second pick, and what was the probability of that happening?
22. Suppose Utah gets lucky and wins the top pick. (a) What is the probability Boston will
win the second? (b) Which team is most likely to win the second pick if Utah wins the first pick?
© Did Utah get lucky at all?
5. Practice Counting
23. Applebee’s has a 3-course meal special with 6 starters, 7 entrees, and 5
dessert choices. How many different meals are possible?
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24. (a) How many different 5-card poker hands are there using a standard deck of
52 cards? (b) How many of these hands contain 4 aces? (c) What is the probability of
getting a 5-card hand with four aces?
25. How many different license plates could there be with 3 letters followed by 4 single
digits from (0-9)? Repetition is possible.
26. Mellow Mushroom in Asheville, NC, has 43 different toppings available for their regular
pizzas. How many different pizzas are possible with (a) one topping (b) two different toppings (c)
three different toppings (d) four different toppings?
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