Ok, you geeks, I ran Chad Ford's ESPN Lottery 100 times. I did this after doing a few times for fun and noticing what I thought was a bias. There are interesting results, or are there?
Wolves occurrences expected
#1 37 25.0
#2 24 21.5
#3 17 17.8
#4 22 35.7
For the statisticians or researchers out there, the two questions I had prior to the simulation were a) does it give the #1 pick to the #1 seed too often, and b) is the distribution as it should be?. A post-hoc question is c) whether there is a bias just with regard to the first and fourth picks. I mean, if something funny is going on, could it be consistently a #1 instead of #4 scenario?
So let's do the easy one, (a)
What is the probability of getting 37 or more occurrences of pick #1 in 100 trials, given the chance of getting the pick is 25% and assuming the trials are independent. If that probability is small, then we don't believe the ESPN lottery is random. For the geekiest among us, my pilot study indicated a one-way test, even though I agree this screams two-way in most situations. The question in my head, though, was why I was seeing the Wolves get the #1 so often. The simulation was done to see if it was biased one way.
Without going into the theory of this, to answer the question and get my final probability, I will sum the probabilities of 37, 38, 39,...,100 to get my answer. If that seems like it's going to be a high probability to some non-geeks, note that at some point the individual probabilities are going to be essentially 0. So, here it is:
The chance of 37 or more #1's when the chance of #1 is 25% is...drumroll... 0.518%. So there is a 0.5% chance (1 in 200) of getting the result we got if ESPN's lottery works the way it should. Conclusion? Well, it's not good to think in black and white and draw a line at an arbitrary % that is acceptable. Often people want to use 5%: if it's less than 5%, it's really improbable that it's working right. We don't want to think like that, but wow, 0.5% is an extremely small chance, and there's no denying that something is wrong. It might be that the trials are not independent. I tried it once, got the Wolves, and maybe there's something that sticks mostly there for the first 100 trials. Maybe it's biased. After all, I believe ESPN knows I'm in Minneapolis. More likely it's biased toward whomever is the 1 seed. Because of #2 and #3 coming up about the same frequencies with which they should, we've really answered (c) without having to do another experiment.
Now for (b). Here we have a multinomial distribution with the value and their probabilities being the "expected" divided by 100. If we determine that the counts are too far away from their expected counts, we can conclude the ESPN lottery isn't working right. My guess is that the probability of the seeing what I saw is about 1%, so let's run it.
I used a chi-square test to determine the probability of deviating this much is 1.00%. Most statisticians will understand why that was intuitive. Note that it is not double 0.518, but it is close. At any rate, the ESPN lottery should not be depended upon to give you proper results. Basically we answered this question with part (a).
So three years ago I'm sitting in a bar in Denver waiting for the lottery to begin. Minnesota had tied for the third worst record in the league and had won the coin flip with Memphis for the third seed. Howver, the number of lottery balls was the average of the third and fourth position, with the Wolves getting the extra ping pong ball, 138 of the 1000 chances (13.8%). Miami had a 25.0% chance, and Seattle had a 19.9% chance. And maybe, just maybe, it would be helpful to know that Chicago had a 1.7% chance. We all knew that chance wasn't really 1.7%, but let's pretend we believe the commish.
So they start unveiling the slots, and we soon learn that Chicago was top three. However, it went true to form after that until the TWolves got the #4--oops!!! They didn't. They, for only the second time ever, had their number drawn at some point. Now, it was widely considered a two-player draft (Rose, Beasley), and Chicago was in there, so it was pretty apparent that the order would be Chicago, Miami, Minnesota...100%. But right there and then, I decided to suspend my cynicism (or truth, as I call it) and pretend there was actually randomness to the draft. The question I had: given that these three teams were in the top three, what is the probability each has each choice? Or more immediately, what is the probability the Wolves have the #1 and Derrick Rose?
So while they went to commercial after announcing Seattle as the #4 pick, I got a napkin and started doing the math. I was both surprised (because I'd never considered this before) and not surprised by the answer. Given the Wolves make the top three and it isn't rigged and the Wolves aren't cursed, what is the probability they get the #1 pick, the #2 pick, the #3 pick? Remember, Miami had 250 chances, the Wolves had 138 chances, and the Bulls had 17 chances. I'll answer below or acknowledge the answer. Unfortunately the order was already set due to the Bulls being a part of this and the Wolves being a part of this, as noted in the previous paragraph...but let's pretend it wasn't.
Back to the ESPN Lottery. It was interesting to me where Chad Ford had placed the Wolves interest. He clearly ranked the Wolves' choices as 1. Irving, 2. Barnes, 3. Williams, and 4. Kanter. No scenario came up to give the Wolves a different player. Only two scenarios I saw had the Wolves taking Williams. These were:
When Utah or Cleveland got the second pick with Irving gone, they took Barnes. BTW, Ford has the Wizards and Warriors taking Williams with the first pick. I'm not sure who the Bucks take with the first pick, but they pass on Irving the time I saw them at #2 (Wizards were #1).