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It's highly likely that you've spent some time recently doing one of the zillion mock lottery things online, or you've at the very least read a few articles about the Celtics' chance of getting off the Tankmobile at "Riggin' for Wiggins" or "Sorry for Jabari" stations. However, without a proper knowledge of how the lottery works, or even without prior exposure to statistics/probability, those numbers can offer you confusion, disappointment or false hopes among many other feelings. I hope that this post will be of help in that department, yet it is also likely that I will confuse you even more, so read at your own risk.

How does the lottery work?

This is something that you would not be able to guess without explicitly reading about it. I had thought that there were a thousand balls with each team's name, logo or some associated number; but that's not even close. There are 14 balls, and teams are allocated possible combinations of those balls. Then someone draws 4 balls out of that bag, and since it is a "combination without repetition", there are 14!/(10!*4!) = 1001 possible combinations. If the combination belongs to a team whose combination wasn't picked previously, they win. The "11-12-13-14" combination is not given to any team, so that leaves 1000 to be allocated among each team.

Now, after the three lottery winning teams are determined, aka the top-3 picks, the remaining teams get their picks according to how much they sucked during the regular season. That's why it's mathematically impossible for the Celtics to pick the 4th (if their combination is picked, they are in top 3, and if not, the highest they can pick is the 5th)

Statistical independence

Now that we've covered the basics, let's move on. I have often heard people say that "Well, the 3rd team wins the lottery the most" or "the last time the worst team won the lottery was in..." etc. Those are definitely correct statements (if one can read Wikipedia properly), yet they are utterly meaningless. Every NBA draft lottery is independent. Yes, in the long run, the ratio of each team being picked will converge to the odds, but again, it's the ratio, so even if this was the 123903218903rd NBA draft lottery, we still wouldn't have been able to infer anything.

Let's say you're tossing a proper coin for the 100th time: If previously you had 50 heads and 49 tails, what would you pick? Tails? Well, it doesn't matter, the odds are still 50/50. Let's say you're tossing it for 1000000th time (why the hell would you do that, don't you feel sorry for your thumb?!?), and you had 500000 heads and 499999 tails, should you pick tails? No. It still doesn't matter. As a matter of fact, when the numbers get large, because it is the ratio that converges and not the exact number of time it's heads or tails, the absolute value of the difference between heads and tails tend to increase. Mind blowing, right?

So yeah, long story short, the past doesn't mean anything. Big numbers are weird.

What do these odds mean then?


Well, as I have previously explained, the lottery is drawn to determine the top-3 picks. Because the Celtics and the Jazz share their odds, they have almost the same number of odds for each pick. The chances of the Celtics getting the top pick is (the number of combinations they have/total number of combinations) = 103/1000 = 10.3%. Easy, right? Now, the odds of the Celtics getting the second pick is tricky though. How would the Celtics get that? Well, they have to a) not get the first pick, and then b) be the second winner of the lottery. The probability of (a) is easy to calculate: 897/1000. But what about (b)? Here's the most important part: The odds of the Celtics getting the second pick is conditional on whoever wins the lottery, and that number you see in the charts, 11.1%, is sorta the average of all those different conditions. Let me clarify: If the Bucks get the first pick, the probability of the Celtics getting the second pick is 12.3%. If the Suns win the lottery, the probability of the Celtics getting the second pick is 9.2%. I hope that makes sense.

So the odds of the Celtics getting the 5th pick is actually an average of all the cases where the top 3 pickers are any three of the Bucks, the 76ers, the Magic, and the Jazz. And remember, on lottery night, the odds will keep on changing.

The highest number in the Celtics' probability row is at the 6th pick. Are we doomed?

Nope, not necessarily. As I have tried to explain above, that chart is the expected probability of given all possible cases. Are you upset that the chances of the Celtics getting the 6th pick is 34.2%? Well, that is still lower than the Bucks getting the 4th pick (35.8%), for example. The likeliest pick isn't higher than the 4th for any team, that's the whole point of this weighted lottery system: It doesn't outrageously favor any team.

I guess what I'm trying to say is: Don't look at a single combination and despair, or if you want to do that, study the odds for every other team too and find your silver lining.

Epilogue:

So, have I confused you even more, or was this post actually helpful to you? Sound off in the comments section, and if you have any questions, I'll be there to answer them.

Oh, and until May 20th, stay flaccid for Embiid.
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semioticus (shelbyl) 4/23/2014 07:32:00 PM Edit
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