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What are the Odds?

The NBA Draft Lottery takes place today at 5 PM Pacific Time.  Not only is it an important event in the history of at least one if not fourteen NBA franchises, it's also the best real-world example of probability that any basketball fan is likely to come across.  So given that I, for some unknown reason, tend to enjoy delving into the minutiae of various subjects, let's talk probability.

(By the way - in case you haven't figured it out, I'm a generalist.  Other than the Clippers, I don't consider myself an expert on anything.  But I know a little bit about a lot of subjects.  As such, if there are any real probability experts out there, please use the comments to add any salient information and/or correct any mistakes I make.)

First, a fun example that I've always found useful in understanding the big picture in probability.  Imagine you are a contestant on "Let's Make a Deal" (by the way, you're in a gorilla suit).  Behind one of three doors is a fabulous prize, let's say it's a car.  Monte Hall tells you to pick a door.  You pick door number 2.  Monte now shows you what is behind door number 1 - it's a year's supply of Spam.  And here's the big question - Monte asks if you would like to change your choice - do you want to stick with door number 2, or change to door number 3?

What should you do?  What is the probability that the car is behind door number 2?  What is the probability that it is behind door number 3?

Most people intuitively think that the odds are the same - that there's a 1 in 3 chance that the car is behind each of the doors.  Believing that, they feel a loyalty to their first choice, and they stick with it.  WRONG!

If you are ever on "Let's Make a Deal" and you find yourself in this situation, ALWAYS ALWAYS ALWAYS switch your choice.  Here's why.  The important factor to bear in mind is not the odds that you made the CORRECT choice originally, but the odds that you made the INCORRECT choice originally.  There's a 2 in 3 chance that you made the incorrect choice.  When Monte showed you the Spam, it now became a 2 in 3 chance that the car is behind door number 3, while there's still only a 1 in 3 chance that it's behind your original door, door number 1. 

What does this have to do with the NBA lottery?  Well, nothing, really.  David Stern is not going to move any further down the game show path than he already has.  But I find that this example helps me understand the subject of probability.  Plus, it has a gorilla suit in it.

We discuss the ping pong balls a lot, and jokingly say that each loss represents more ping pong balls.  But that's not actually true.  In fact, there are only 14 ping pong balls, each with a number from 1 to 14.  Each non-playoff team is then, according to their order of finish on the regular season, assigned a certain number of combinations of those numbers.

There are 1001 possible combinations of 14 ping pong balls, without replacement and without regard to the order.  OK, what does that mean?  Without replacement means that the ping pong ball with the number 7 on it can only appear once - 7-7-7-7 is not a valid combination.  With replacement, the numerator of your equation would be 14 to the fourth power - 14 times 14 times 14 times 14.  Without replacement, the value goes down by one each time - 14 times 13 times 12 times 11.  Makes sense.

But that's a really big number - it comes out to 24,024 in fact.  That's how many combinations there would be if the order mattered: if 1-2-3-4 were considered a different possibility than 1-2-4-3.  But order does not matter in the case of the NBA lottery.  When order does not matter, you divide the numerator by 4 factorial [notated (4!)], or 4 times 3 times 2 times 1 (4 being the number of balls that will be chosen).  This combinatorics problem would commonly be called 14 choose 4.

So, the NBA draft lottery is really a case of 14 choose 4, which gives us:


24024/24 = 1001

(I think the real equation is actually (14!)/(4!)*((14-4)!) but the (10!) in the numerator and in the denominator cancel out, leaving the above equation.)

The NBA takes those 1001 combinations and assigns 1000 of them to the 14 lottery teams.  The last one is unassigned, and in the 0.1% chance that that combination comes up, they'll just redraw. 

The Clippers have 177 combinations assigned to them.  Remember that LA lost a coin flip to the Wizards, who consequently got one extra chance, or 178.  That gives LA a 17.7% chance of winning the first pick when the four ping pong balls are drawn.  That one's pretty straightforward - 177 chances in 1000 makes 17.7%.

The math for the second and third picks gets more complex.  I don't know all of the details, but I know in general how it works out.  The Clippers odds of getting the second pick is 17.29%, and for the third pick they have a 16.41% chance.  You may be wondering, if their 177 combinations are all still available (which makes sense, given that they must not have gotten the first pick), why are their odds going down?  Here's where the "Let's Make a Deal" example is useful.

Remember to consider the odds that they DIDN'T win the first pick just like you have to remember the odds that you didn't pick the right door when Monte asked.  If they have a 17.7% chance of winning the first pick, they have an 82.3% chance of NOT winning it.  You have to multiply that 82.3% by their odds of winning the second pick, since that's the situation in which they would be eligible for the second pick.  So 82.3% of 17.7% is 14.5671%.  Ooops.  Wait a minute.  Now the odds went down too much, since the NBA tells us that the Clippers have a 17.29% chance at the second pick. 

Here's where it gets complex.  The Clippers still have 177 combinations in play - but it's not longer out of 1000.  Whoever won the first pick is now out of contention, taking all of their combinations with them (those combination become just like the 1001st combo, a do over if they happen).  But the math gets nasty, because we don't know which team will win the first pick.  If the Kings win, then 250 combinations go away, and the Clippers have 177 chances in 750 (23.6%) - but if the Suns win, then only 5 go away, and the Clippers have 177 chances in 995 (17.8%).  The NBA super computers in Secaucus calculated all of the possibilities, and came up with that 17.29% chance the Clippers get the second pick.  I'm just going to believe them on that one.

So there you have it.  17.7% chance they get Blake Griffin, and 17.29% chance they get Ricky Rubio.  Not bad, right?

This is a cruel, cruel year to be in the lottery.  Because that 35% chance of winding up with one of the two consensus prizes in this draft, is up against the 65% chance that they DON'T get a top two pick.  They can't drop below 6, but in this draft, who cares if you're picking third or sixth?  Hell, in some ways third is worse - sure, you get your pick of everyone not named Griffin or Rubio, but that just adds to the pressure, not to mention making the disappointment that much greater if the pick doesn't pan out, which let's face it, could definitely happen.

Will a player from this draft other than Griffin and Rubio turn into a great pro?  Undoutedly.  There is always hidden talent in every draft; the trick is finding it.  The last two "Two player" drafts were 2004 (Dwight Howard and Emeka Okafor) and 2002 (Yao Ming and Jay Williams).  Obviously, in retrospect, those weren't even the right two players (although in Williams' case it was unforeseeable that his career would end in a motorcycle accident).  Howard and Yao are now the two best centers in the NBA, but Amare Stoudamire (9th in 2002), Tayshaun Prince (23rd in 2002), Carlos Boozer (34th in 2002), Devin Harris (5th in 2004), Andre Iguodala (9th in 2004), Al Jefferson (15th in 2004), Jameer Nelson (20th in 2004) and Kevin Martin (26th in 2004) are among the players to come out of those "two player" drafts.  But there were also a lot of bad lottery picks those years. 

At any rate, this is an important lottery.  I'll have an open thread up at 4 PM dedicated to the lottery.  And of course you discuss your thoughts on this thread until then.  If the Clippers are picking first or second in June, the prospects for the near future improve markedly.  There's a 35% chance that will happen.  The other 65%?  Odds are, it doesn't change much.