Lottery Logic: A Logical Look At A Lucky Lottery By now, most NBA fans are well aware of the fact that the Bucks finished the 2006-07 NBA season with the third-worst record, thus earning themselves the third-most combinations for the May 22 NBA Draft Lottery. Most naturally assume (as did we) that this means Milwaukee is more than likely going to land the third pick in the draft. But we were surprised to find out that we were mistaken (something that rarely happens). In fact, statistically speaking, the Bucks are favored to land the fifth pick in the draft.
We know. It didn't make sense to us either. So we went outside of our cubes and found someone really smart who could explain it to us. Enter Robert (or Bob) Collins, Adjunct Assistant Professor at Marquette University's College of Business Administration. He's the guy pictured to the right and he's been teaching math at MU since 1987. We told Bob of our confusion and he said he would be happy to break it down for us. His explanation can be found below and it actually makes some sense. But despite the odds, we're still hoping for number one! Enjoy Bob's handy work: How is it possible that the Bucks had the third worst record in the NBA, yet their most probable lottery pick is the fifth? There are a few ways to try to answer this. Surprisingly, if a Grizzlies fan looked at the same information, they would notice (not happily) that the fourth pick is the most likely for Memphis. A Celtics fan would also see the fourth pick is the most likely draft spot for Boston. How does this happen? If you look at the probabilities for the first pick, Memphis has the highest probability at 25%, meaning there is a 75% chance someone else will get the first pick. While Memphis not getting the first pick will make one other team extremely happy, it hurts the chances of everyone else getting the second pick as Memphis will still have the most 'ping pong balls' in the lottery for the second pick. If Memphis does not get the first pick, the probability they get the second is between 25.1% (if the Clippers get the first pick) and 31.2% (if the Celtics get the first pick). Since Memphis has the most 'ping pong balls', they become the team with the highest probability to get the first pick, second pick, and third pick. Since there is only a 64.3% chance they get one of the first three picks, it turns out they also have the highest probability of getting the fourth pick. As long as Memphis is in the lottery, they will be the most likely team for the next pick, followed by Boston, the Bucks, Atlanta, and so on. Within each round, the order of the probabilities for the remaining teams makes sense. Another way to try to explain how the fourth pick is so likely for Memphis, Boston, and the Bucks is to look at the probability someone else wins one of the top three picks. There is a 90.2% chance that at least one team other than Memphis, Boston, and the Bucks will get one of the top three picks. Once this happens, at least one of these teams will get a lower pick. The last way to try to explain the probabilities is to look at every possible draft order. Trust me, this will hurt (see the Excel spreadsheet). When the NBA holds its draft lottery, there are 2184 (not equally likely) different possibilities for the order of the first three picks. Given that many possibilities, no one draft order will have a very large probability. The most probable order (Memphis, Boston, Milwaukee) has a probability of only 1.9%. The least probable order (LA Clippers, New Orleans, Philadelphia) has a probability of 0.00002%. Below is an explanation of the probabilities associated with each of the possible Bucks draft positions. The Bucks draft first: Since the Bucks have 156 of the 1000 'ping pong balls', the probability of the Bucks getting the first pick is 15.6%. If one looks at the 2184 possibilities for the draft order, 156 (a coincidence that this is also the number of 'ping pong balls' allotted to the Bucks) have the Bucks first in the draft order. The Bucks draft second: There are 156 possibilities that lead to the Bucks drafting second. The sum of the probabilities of these 156 possibilities is 15.73%. The most probable order with the Bucks drafting second is Memphis, Milwaukee, Boston with a probability of 1.7%. The Bucks draft third: There are 156 possibilities that lead to the Bucks drafting third. The most likely (Memphis, Boston, Milwaukee) has a probability of 1.9%. The Bucks draft fourth: There are only 66 possibilities that lead to the Bucks drafting fourth, but many have (comparatively) high probabilities. The sum of these probabilities is 22.55%. For the Bucks to draft fourth, both Memphis and Boston would have to be in the first three positions (which is quite reasonable) with someone other than the Bucks getting the other position. The most likely order leading to the Bucks getting the fourth pick is Memphis, Boston, Atlanta with a probability of 1.4%. The Bucks draft fifth: There are 660 possibilities leading to the Bucks drafting fifth, giving a probability of 26.49%. The most likely order leading to the Bucks drafting fifth is Memphis, Atlanta, Seattle at 0.55%. There is not a very high probability of any single possibility leading the Bucks drafting fifth, but the large number of possibilities leads us to the sum of 26.49%. The Bucks draft sixth: There are 990 possibilities leading to the Bucks drafting sixth, but the probability of each is very, very small, so the sum is only 4.05%. The only way the Bucks can draft sixth is if Memphis, Boston, and the Bucks are all shut out of the top three picks. The most likely possibility here is Atlanta, Seattle, Portland with a probability of 0.08%. |
