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# Hornets Lottery Probabilities 3

*Do we know our chances yet?*

Not quite yet. We’ll update live from Jazz Fest when the last wrinkles are ironed out. Yeah, that’s the kind of ship we run around here. Beach Boys reunite, fist full of couchon de lait, all while updating Hornets fans world-wide about coin flips a thousand miles away. Bienvenue a New Orleans.

*Update: The Cavaliers won the tie-breaker.*

In the first and second posts about the New Orleans Hornets’ lottery probabilities, there was uncertainty about our draft position. Now, that uncertainty is eliminated, replaced by a single flip of the coin, which will be performed later today. We present an in-depth look at the lottery in general, a look at how each pick of the Hornets may deliver, then a look at the combination of the picks.

Hornets247 will have more coverage in the coming weeks, of course.

**NBA Draft Lottery**

*The Short:* The 14 teams that do not make the NBA playoffs enter the NBA Draft Lottery. The first three draft picks are doled out via the lottery procedure. The remaining picks are assigned based on the initial slotting.

*The Long:* The NBA Draft Lottery is used to assign the top three draft picks in the NBA Draft. The 14 teams not in the NBA Playoffs are entered into the lottery. Teams are ranked in terms of number of wins, with the team with lowest wins being in slot 1, and the team with the most wins being in slot 14. We discuss ties below, but assume that there are no ties for now.

14 lottery balls, which seem like ping-pong balls with a number printed on them in some pattern, are placed into a randomizing hopper. Four lottery balls are drawn without replacement. The combination, without respect to order, is one draw in the lottery. This combination appears in exactly one of lists of combinations assigned to each team in the lottery. The number of combinations in each list is different for each team, with more combinations being assigned to teams with fewer wins. The particular number of combinations is discussed below.

There are 1001 possible draws from the lottery. One of these combinations is ignored if drawn at any time. In this case, the lottery balls are re-entered into the hopper, and the officials draw again. After the first draw of the lottery that corresponds to a team, the combinations for that team are designated as redraw combinations, and that team is awarded the first pick in the NBA Draft. The procedure is repeated until the first three lottery picks are assigned, with the picks being assigned in order.

In the event teams are tied in groups, the entire list of teams is ordered by wins with tied team being listed arbitrarily within those constraints. A `coin flip’ is then executed which ranks those tied teams. Tied teams are then awarded the average number of combinations assigned to teams in the slots they occupy based on the untied condition. If the average number of combinations for those tied teams is not an integer, the number of combinations equal to the greatest integer less than the average is assigned to each team. The remaining combinations are awarded to the team ranked most highly by the random procedure.

The ties not only affect the teams that are tied, but they also affect the chances other teams have at the second and third picks (the first pick probabilities are immune to these effects). This is because of the way teams that are selected for the first two picks have their combinations effectively removed from the lottery in succession.

The NBA Draft Lottery will be held on May 30, 2012, and the coin flips, as mentioned above, will be later today.

*Update: The Cavaliers won the tie-breaker.*

**Hornets Slot**

The Hornets are tied in the 3rd slot. These teams are assigned 137 or 138 of the 1000 combinations, with the 138 assigned the winner of the coin flip. Like all slots, any of the top 3 picks can be awarded to the team in this slot, along with picks 4, 5, and 6 if the Hornets win the coin flip and 4,5,6,7 otherwise. No other picks are possible.

The following table lists the probabilities of getting these picks from this slot in this draft to four digit precision if we win the coin flip. Also, note that here and elsewhere, round off error may frustrate attempts to get certain sums to be equal. Rest assured, these calculations were performed using sufficient precision to accurately report the probabilities to four digits.

Pick | Probability |
---|---|

1 | 0.1380 |

2 | 0.1424 |

3 | 0.1453 |

4 | 0.2382 |

5 | 0.2905 |

6 | 0.0455 |

And if we lose the flip *(we did)*:

Pick | Probability |
---|---|

1 | 0.1370 |

2 | 0.1416 |

3 | 0.1447 |

4 | 0.0851 |

5 | 0.3231 |

6 | 0.1558 |

7 | 0.0127 |

**Timberwolves Slot**

The Timberwolves are untied in the tenth slot. This slot is assigned 11 of the 1000 combinations. Like all slots, any of the top 3 picks can be awarded to the team in this slot, along with picks 10, 11, 12, and 13. No other picks are possible.

The following table shows the probability of getting these picks from this slot in this draft to four digit precision.

Pick | Probability |
---|---|

1 | 0.0110 |

2 | 0.0130 |

3 | 0.0157 |

10 | 0.8699 |

11 | 0.0886 |

12 | 0.0018 |

13 | 0.0000 |

The 0.0000 entry indicates a non zero entry whose probability is below 0.00005.

**Lottery Outcomes**

The following table shows the nonzero probabilities of each nonzero draft outcome to four digit precision in the event we win the coin flip. As above, if the outcome does not appear, it has zero probability.

2 | 3 | 4 | 5 | 6 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|

1 | 0.0033 | 0.0037 | 0.0014 | 0.0049 | 0.0016 | 0.1250 | 0.0090 | 0.0001 | |

2 | 0.0042 | 0.0019 | 0.0058 | 0.0017 | 0.1298 | 0.0087 | 0.0001 | ||

3 | 0.0026 | 0.0069 | 0.0018 | 0.1338 | 0.0080 | 0.0001 | |||

4 | 0.2185 | 0.0138 | |||||||

5 | 0.2331 | 0.0389 | 0.0009 | ||||||

6 | 0.0296 | 0.0102 | 0.0006 | 0.0000 |

To read this chart, you must have a pair of picks in mind. The cell at the intersection of the row of the better pick and the column of the worse pick contains the probability of that pick combination happening, regardless of how they arise. For instance, there is a 0.0058 probability of the Hornets receiving the second and fifth picks in the draft if they win the coin flip.

The following chart is similar to that above, but contains the probabilities if the Hornets lose the coin toss *(we did)*.

2 | 3 | 5 | 6 | 7 | 10 | 11 | 12 | 13 | |
---|---|---|---|---|---|---|---|---|---|

1 | 0.0033 | 0.0037 | 0.0032 | 0.0041 | 0.0007 | 0.1241 | 0.0089 | 0.0001 | |

2 | 0.0041 | 0.0039 | 0.0047 | 0.0007 | 0.1291 | 0.0086 | 0.0001 | ||

3 | 0.0052 | 0.0054 | 0.0007 | 0.1332 | 0.0080 | 0.0001 | |||

4 | 0.0851 | ||||||||

5 | 0.2818 | 0.0290 | |||||||

6 | 0.1102 | 0.0303 | 0.0012 | ||||||

7 | 0.0064 | 0.0038 | 0.0004 | 0.0000 |

Inspection shows us that the Hornets most likely single outcome is to pick fifth and tenth, regardless of result of the coin flip. In each case, this outcome happens about 1/4 of the time.

These tables show all the possibilities and probabilities, but they aren’t easy to use, so we have some summary tables to help answer the popular questions.

*What is the probability we get the top pick and at least a top X pick to go with it?*

The first table shows the probabilities if we win the coin flip, the second if we lose the coin flip *(we did)*.

X | Probability |
---|---|

2 | 0.0033 |

3 | 0.0070 |

4 | 0.0085 |

5 | 0.0134 |

6 | 0.0149 |

10 | 0.1399 |

11 | 0.1489 |

12 | 0.1490 |

X | Probability |
---|---|

2 | 0.0033 |

3 | 0.0070 |

5 | 0.0102 |

6 | 0.0142 |

7 | 0.0149 |

10 | 0.1390 |

11 | 0.1479 |

12 | 0.1480 |

The probability next to the pick indicates the probability that we receive the top pick and at a top X pick. So, there is a probability of 0.0134 getting the top pick and a top 5 pick to go with it if we win the coin flip.

Also, a `missing’ X has the same probability as the largest X that is listed that is also smaller than it. So the probability that we get the top pick and a top 13 pick to go with it if we win the coin flip is that same as that for the top pick and a top 12 pick if we win the coin flip. This convention is used in the following as well.

*What is the probability we get at least a top X pick?*

The first table shows the probabilities if we win the coin flip, the second if we lose the coin flip *(we did)*.

X | Probability |
---|---|

1 | 0.1490 |

2 | 0.3011 |

3 | 0.4543 |

4 | 0.6866 |

5 | 0.9595 |

6 | 1 |

X | Probability |
---|---|

1 | 0.1480 |

2 | 0.2993 |

3 | 0.4519 |

4 | 0.5370 |

5 | 0.8478 |

6 | 0.9894 |

7 | 1 |

For example, if we lose the coin flip, there is a 0.5370 probability of having a top four pick in the draft.

*What is the probability both picks are in the top X?*

The first table shows the probabilities if we win the coin flip, the second if we lose the coin flip *(we did)*.

X | Probability |
---|---|

2 | 0.0033 |

3 | 0.0112 |

4 | 0.0171 |

5 | 0.0346 |

6 | 0.0397 |

X | Probability |
---|---|

2 | 0.0033 |

3 | 0.0111 |

5 | 0.0234 |

6 | 0.0376 |

7 | 0.0397 |

For example, if we win the coin flip, there is a 0.0112 probability of having both of our picks in the top three of the draft.

If you have specific questions, put them in the comments and we’ll address them.

Enjoy.

*Update:* A question arose as to how likely it was the coin flip we lost would affect us against the Cavaliers. From a talent perspective, we should be targeting different players if outside the top three, according to Mike. From a probabilistic perspective, our odds at getting into the top three are about even with them regardless. From a tie-break perspective, this will come into play with a probability of 0.2892, the probability that both the Hornets and Cavaliers miss the top three.

Bee dat

April 27, 2012 at 8:03 am

15% chance of pure unadulterated bliss! still dreaming pf adavis and kmarshall!

Bee dat

April 27, 2012 at 8:05 am

look forward to the day when the juices from that cochon can dribble down onto the previous night’s fan-up shirt!

Mason Ginsberg

April 27, 2012 at 8:05 am

Fantastic work Jason. This is great info. Well done.

Joe P

April 27, 2012 at 10:53 pm

yeah, seriously. amaz.

David

April 27, 2012 at 9:03 am

That’s just dumb. My brain hurts :/

nikkoewan

April 27, 2012 at 9:03 am

You stole my work!!!! But seriously, cool post. It’s exactly what I had in mind. Good job!!!

Dee

April 27, 2012 at 9:27 am

I’ll just wait until the lottery to understand

edbballin504

April 27, 2012 at 9:34 am

As a finance degree recipient, I have to say that was some great work Jason! Well done! You make it as easy as possible to understand. If anyone is lost and confused shame on them.

Jason Calmes

April 27, 2012 at 11:19 am

Thanks, Ed. I thought “ballin” was a basketball thing. . . with that finance background, I think you are talking about ducats, man. Buckets o’ ducats!

Next round is on Ed, people!

Mason Ginsberg

April 27, 2012 at 9:43 am

The most amazing thing to me (which Jason and I are currently discussing) is the incredible variance for the 4th & 10th pick combination based on the outcome of the coin flip with Cleveland. 22% chance for 4+10 if we win it, but that falls all the way to 8.5% if we lose it. Crazy difference!

Michael McNamara

April 27, 2012 at 9:51 am

That’s because, if we lose coin toss, Wash, Charlotte, and Cleveland would all have to get their balls to come up for us to stay at 4, while if we win coin toss, one team could come up from behind and knock us back a spot. It is rare, if ever, that teams in 4, 5, 6 positions stay in their slot

Mason Ginsberg

April 27, 2012 at 9:54 am

Yeah, we talked about it. Makes complete sense, just crazy to see the difference on paper.

Jason Calmes

April 27, 2012 at 10:12 am

You are exactly right, Mike, but when I saw the result just in isolation, it was sphincter-tightening, as I told Mason. Very stark example of the dependent sampling. Also, great example of the power of fault-tolerant systems if viewed as a reliability problem.

Fun little problem.

504ever

April 28, 2012 at 1:16 am

I hope we get lucky and pick 1, 2 or 3. Then losing the coinflip is minimized, and we get to pick before the second significant talent drop off that I see from pick 3 to pick 4. (The first talent drop off id from Davis to pick #2).

champsworld504

April 27, 2012 at 10:24 am

Stern Magic will land us #1 and #2. espn analysts will commit suicide.

Rocco

April 27, 2012 at 10:58 am

Jason:

You’ve missed your calling. THe IRS is looking for talent like yours to write their forms instructional manuals. 🙂

Very thorough.

Jason Calmes

April 27, 2012 at 11:10 am

Did you just invoke the IRS in a compliment? Is that even possible?

I’m going to hop on my perpetual motion bike and head to the fountain of youth while my frige cools my house.

David

April 27, 2012 at 11:00 am

So if we’re tied for the 3rd worst record, we should have the best odds for the 3rd pick and 2nd best for the 4th pick, correct???

Jason Calmes

April 27, 2012 at 11:08 am

Nope. We are most likely to pick 5th with out pick.

In untied cases:

Slots 1 and 2 are most likely to pick fourth.

3 and 4 are most likely pick fifth (so tied doesn’t mess with this really).

5 is most likely to pick sixth.

6 and higher are most likely to pick according to their slot.

Edited to add: This is counter-intuitive. No team’s most likely outcome is getting any of the top 3 picks. Teams are more likely than others to get it, and one must, but no team is most likely to hit a particular lottery pick. Only the top 2 slots are more likely to end up in the top 3 than out of it, assuming one pick, no ties, etc.

Chuck

April 27, 2012 at 12:19 pm

Great post, even I have no criticisms

I still think that Minny pick is winning the lottery just so Bill Simmons can type KAHNNNNNNNNNNNNNNNNN one last time

Jason Calmes

April 27, 2012 at 1:11 pm

It’s morning in America!

Good stuff.

J

April 27, 2012 at 12:23 pm

They need to end the lottery

Lucas Ottoni

April 27, 2012 at 12:40 pm

My order of selection (with our first pick):

ANTHONY DAVIS

ANDRE DRUMMOND

THOMAS ROBINSON

MICHAEL KIDD-GILCHRIST (Ariza will be traded)

My order of selection (with our second pick):

KENDALL MARSHALL

DAMIAN LILLARD

JARED SULLINGER

TYLER ZELLER

cloud520

April 27, 2012 at 12:48 pm

My Anthony Davis pipe dream lives!

edbballin504

April 27, 2012 at 2:24 pm

So how do we get the results of the coin flip?? When? Where?

Jason Calmes

April 27, 2012 at 2:48 pm

It’s in the post, edbskimmin.

Cavs won. I updated it an hour ago (Thanks, Mike).

edbballin504

April 27, 2012 at 2:54 pm

I just noticed the update! That sucks! lol. Hope that pointless win vs GSW was worth it to Monty now! Now the long wait begins! May 30!

Jason Calmes

April 27, 2012 at 3:05 pm

It’s all odds. Look at it as bonus.

We’d be in anotger tie if we lost that one if Washington didn’t tank harder as a result.

Jake Madison

April 27, 2012 at 3:30 pm

Couchon is awesome.

kempleton

April 27, 2012 at 3:32 pm

Well well!

It is time to start countdowns:

Countdown #1 for the Draft lottery: T – 33 days and counting!

Countdown #2 for 2012 NBA Draft: T – 62 days and counting!

Here are some important days:

May 30 2012 NBA Draft Lottery

June 6-10 2012 NBA Draft Combine

June 28 2012 NBA Draft

July 11 NBA Teams May Begin Signing Free Agents

Geaux Hornets!

Icebird

April 27, 2012 at 5:11 pm

Looking the probabilities again, losing the coin-flip hasn’t changed the Hornets probabilities of getting a top three pick all that much, or of winning the Antonio Davis sweepstakes. It’s just shifted the “most likely results” from the 4th or 5th pick, to the 5th or 6th pick. I could probably dig out the analysis that’s been done in previous years on how picks in that range pan out, but I suspect there isn’t a large difference between say number 4 picks and number 6 picks over the long term. Each slot probably has a couple of stars, a larger number of players who might be good but not team-changers, and probably a bunch who never panned out at all.

Zombian

April 27, 2012 at 6:01 pm

Antoine Davis.

edbballin504

April 27, 2012 at 5:49 pm

Interesting stat if you’re a Bobcats fan (do Bobcat fans even exist?): In 27 lotteries, only 4 teams have won the NBA lottery after having the worst record that season:

1. 2004 ORL – Dwight Howard

2. 2003 CLE – LeBron James

3. 1990 NJN – Derrick Coleman

4. 1988 LAC – Danny Manning

In other words, don’t get your hopes up CHA fans!

DownUnder

April 27, 2012 at 7:48 pm

Great work on the post, obviously quite some time working all on those numbers.

We’ve gotten all our bad luck out of the way with the coin flip, now its time to win everything else!

Lucas Ottoni

April 27, 2012 at 11:39 pm

I think the teams that received the first pick in the last three years shouldn’t compete again for the first pick. Teams like Wizards and Cavaliers should enter the lottery from the second pick. The NBA could think about it.

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