Lest this become a sports-draft blog, I feel contractually obligated to post about the **NBA Draft Lottery**, what with the Cavs winning and the statistical odds and all.

Last night, the **Cleveland Cavaliers** (my favorite team for you non-twitter followers) won the NBA Draft Lottery, meaning, by chance they were awarded the #1 overall pick in June. This, despite not having the worst record.

For the uninitiated, the NBA Draft is unique among leagues’ player selection drafts in that **the top pick does not necessarily go to the worst team.** You see, in order to prevent teams from “tanking” (that is, losing intentionally at the end of the season in order to obtain a higher draft pick) the NBA has instituted a lottery system. Teams with worse records have higher odds of having a higher pick, via ping-pong ball selections, but there’s not guarantee that the losingest team in the NBA will have the highest pick. The logic is that by not automatically awarding teams picks based on record, that will potentially prevent teams from throwing games at the end of the season. To be sure, it still happens, but probably not to the extent that it would otherwise.

The end result is that **the worse a team does, the better chance they have to pick higher**. And in the NBA, which only allows five players on the court at a time, having a single high pick is generally more valuable than in any other sport.

Now, a bit about the **mechanics**. The Draft Lottery is conducted using **an overly complicated method of ping-pong balls and number combinations**. Ping pong balls numbered 1-14 are drawn at random, yielding a potential 1001 combinations (order is disregarded, that is 1-2-3-4 = 4-3-2-1). Each of the 14 teams that do not make the playoffs are assigned a certain number of number combinations. The worse a team’s record, the more number combinations they are awarded. (Note: the specific number combination of 11-12-13-14 is disregarded, yielding exactly 1000 assigned number combinations)

The team with the worst record is assigned **250 number combinations**, good for a 25% chance at being the first team drawn.

The team with the second-worst record is awarded 199 number combinations (19.9% chance). The team with the third-worst record is awarded 156 (15.6%) and the team with the fourth worst record is assigned 119 (11.9%) combinations. This proceeds all the way down to the team with the highest record of all the non-playoff teams being awarded 5 number combinations, good for an unlikely 0.5% chance of being awarded the first overall pick.

There’s one more catch for the mechanics, then we’ll get to the actual driving question of this: **the lottery is only utilized for the top 3 picks**. After the top 3 picks are determined, the remaining draft picks are awarded based **solely on record**. So, in effect, if you have the worst record in the NBA, you will pick no worse than 4th. And that’s if your number combos (totally 25% of the combinations) are somehow not selected the in first three. Similarly, the second-worst team can slip no further than 5th. And so forth.

[Looks at teacher notes: “Check for understanding”]

Does this make sense?

[Sees lots of confused looks]

Here’s the long and short of it:

- 1: 25%
- 2: 19.9%
- 3: 15.6%
- 4: 11.9%
- etc.

Now, on to the driving question.

**Artifact**

Last night the Cavs were awarded the #1 overall pick. The Minnesota Timberwolves, who had the worst record in the NBA, were **awarded the #2 overall pick**. David Kahn, the General Manager of the Timberwolves went on record, **complaining about how the lottery system is rigged**.

“This league has a habit … of producing some pretty incredible story lines,” Kahn said, while smiling, on Tuesday. “Last year it was Abe Pollin’s widow and this year it was a 14-year-old boy… We were done. I told (Utah executive) Kevin (O’Connor): ‘We’re toast.’ This is not happening for us, and I was right.”

(Last year, they were beat out by a widow, representing the Wizards, this year by the 14-year-old son of the Cavs owner, who has a nerve disorder.)

**Guiding Questions**

- Does Kahn have a point?
- Should the Timberwolves be upset with the second pick, or happy that they didn’t slide all the way to fourth?
- If you have the worst overall record, you have the best odds for the top pick, but that’s only 25%. So what’s your “expected” pick?

**Potential Solution**

It should be noted that the Timberwolves have had especially terrible “luck” with the NBA Draft Lottery system, at least according to them. They’ve been in the lottery 14 times and **never been awarded a higher pick than their record should indicate**. In fact they’ve ended up with a lower pick than their record would indicate 8 times. Some Timberwolves fans says this makes them **0-14** in the lottery. I’d say more like **0-8-6**. That’s a whole other blog post. There’s also the subtext of Cleveland being awarded the top pick a year after losing LeBron James, but whatever. Our concern is with this particular draft.

Namely, if a team has a 25% chance of being selected (via 250 out of 1000 number combinations), **should they expect to get the top pick?** Should they be happy they didn’t get the 4th (or 3rd) pick? If not, what pick should they expect to get, and how often? (side note: mean, median, and mode anyone?)

Obviously, the team with the worst record should expect to pick 1st 25% of the time and either **2nd, 3rd, or 4th 75% of the time**.

So we need to split up that 75%, right? Depending on who gets the #1 pick, that will affect the remaining odds. Last night, for instance, the Cavs were awarded the #1 pick via a trade with the Los Angeles Clippers, which only had a 2.8% chance of hitting. (Interestingly, the Cavs’ original pick was the “biggest loser”, slipping to 4th, despite having the second-most opportunities.)

The Timberwolves started with a 250/1000 chance of hitting, then after the first pick was awarded, if we eliminate the 28 combinations for the Clippers, they had only a 250/972 chance of being awarded the second pick, good for only 26%. 26% of 75% is 18.75%.

So the T-wolves had a 25% to get #1, a 18.75% chance to get #2, and, by subtracting from 100, a **56.25% chance** of ending up with the 3rd or 4th pick.

Had the Cavs won the #1 overall pick with their original pick, with 199 out of 1000 combinations, the T-wolves would have had a 250/801 (31%) chance at the second pick. (for some mental gymnastics, we can say that the T-wolves had a 19.9% chance to have a 31% chance at the second pick)

For general purposes (i.e. no prior knowledge of who the #1 pick is), the NBA Draft Lottery wiki has done the calculations for us:

Seed Chances 1st 2nd 3rd 4th

1 250 .250 .215 .178 .357

According to this, if you have the worst record in the NBA, the odds of you receiving the 3rd or 4th pick, worse than the Timberwolves landed at #2, total to 54%. **The most common (mode) pick the worst team can expect is the worst possible pick at #4**.

With odds like that, **should the Timberwolves consider themselves lucky** to only be picking as low as 2nd?