10.04.07
Posted in Math, Education, Pop Culture at 10:04 am by leingang
Today marks the 50th anniversary of the launch of Sputnik 1, a mostly harmless satellite but only the first man-made object to orbit the earth. It had a profound effect not only on world politics but on education in the United States. As written in Air Force Magazine this month:
In beating Washington to the punch in space, the Kremlin really hit us where it hurt—in our technological ego. Sputnik instantly catapulted the Soviet Union onto the world’s scientific top shelf, raising doubts about America’s own standing…After initial soul-searching, the US embarked on a massive and determined space effort. The Pentagon formulated a huge program. On the civilian side, newly created NASA did the same. The aerospace industry exploded. Colleges were flooded with new engineering students eager to take up the Russian challenge. Public education turned hard toward math and science curricula. Sputnik may have started the Space Age, but America created the Space Race. Soon, the US was to leave Moscow in the dust.
Thanks, comrades! But that was 50 years ago—I’m looking forward to the next big push in science and math education. Or, results of it.
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Tags: sputnik, education
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05.23.07
Posted in Math, News, Pop Culture, Math E-304 at 11:55 am by leingang
The 2007 NBA Draft Lottery was last night and has a nice little probability problem in it. Although it’s pretty painful if you’re a Grizzlies or Celtics fan.
Most professional leagues have a policy of awarding the top picks to the worst teams. But the NBA wanted to discourage tanking a season to get a top pick, and so removed this certainty by instituting a draft lottery to award the top three picks. Teams with worse records are weighted to have a higher probability of winning, but the weights are small enough to make the top pick to the worst team far from certain:
Here’s what actually happened, along with the probability of winning for each team:
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LOTTERY RESULTS
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TEAM
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REC.
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ODDS
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32-50
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5.3%
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31-51
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8.8%
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30-52
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11.9%
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22-60
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25.0%
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|
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24-58
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19.9%
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28-54
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15.6%
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|
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32-50
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5.3%
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33-49
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1.9%
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33-49
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1.9%
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|
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33-49
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1.8%
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35-47
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0.8%
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35-47
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0.7%
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39-43
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0.6%
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40-42
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0.5%
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As you can see, none of the teams with the worst three records (Memphis, Boston, and Milwaukee), and thus the biggest probabilities of winning, ended up with any of the top three picks. Mathematicians often get the FAQ “What are the odds of that?” Well, let’s see.
Let 
be the probability that the team with the ith worst record wins the lottery. The second pick is determined by the same rules, except that the team that got the first pick can’t get the second pick. So the probability that team j (i.e., the team with the jth worst record) gets the second pick, given that team i got the first pick is 
. The third pick is assigned the same way. So the probability that team i gets the first pick, team j gets the second pick, and team k gets the third pick is
All the other teams are assigned picks according to worst record among those remaining. So there’s no more probability to be determined.
From the above you can see that the probability of the lottery ending up precisely this way is f(6,5,4) = 0.00068228, or .068%. Very unlikely. But so is the most likely event that the worst teams gets the first pick, the second-worst the second pick, and the third-worst the third pick. That’s f(1,2,3) = 1.88%. The more interesting event is that none of the worst three teams got any of the top picks. The probability of that is the sum
over all triples (i,j,k) of distinct elements from the set 4, 5, 6, …, 14. There are (11)(10)(9)=990 such triples. There might be a nice combinatorial way to get a closed-form expression for this sum, or you can just use Mathematica to do it. I got 0.0405467, or 4.05%. Not that likely, but you would expect it to happen about once every 25 years. The lottery in its current form has been around since 1990.
Rich Zuckerman at NBC Sports published a table of the likelihood of all the bottom fourteen teams getting any of the lottery picks. The event that no bottom-three team gets a top-three pick is equivalent to the event that the third-worst team gets the sixth pick (think about that for a while). He’s got the same 4.05% figure in that position.
Is the draft lottery a good idea? The math is there, and those who make the rules should be content with the consequences. If having as “unfair” a distribution of picks as this occuring this often is acceptable to the team management, then they should keep it as it is. But the lottery was instituted to discourage teams from “tanking” (not trying to win games) the rest of the season so as to increase their draft position. I’m not sure it does. In a non-lottery system, the team with the worst record is awarded the first pick. In a lottery system, the team with the worst record is awarded the highest probability of the first pick. Among the available choices, that’s still the most attractive, so bad teams still have the incentive to try to have the worst record. But teams who legitimately “earn” the worst record are no more likely to be rewarded with high draft picks. So this system doesn’t reward the honest, or punish the dishonest, it just makes the crime pay less.
(I said earlier that this is particularly painful to Celtics fans. I live in the Boston area, but I am not a Celtics fan. I do sympathize with them, though. My point of view on the lottery is really only based in mathematics and, I guess, game theory.)
.
technorati tags:math, mathE-304, probability, basketball, nba, lottery
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05.22.07
Posted in Math, Funny, Pop Culture at 5:31 am by leingang
Somebody’s uploaded clips from one of my favorite math movies (not a very long list) to Youtube: Donald in Mathmagic Land. It’s a little featurette designed to teach the beauty and usefulness of mathematics. Released in 1959, I have to think that it was part of the space race and math-and-science push spurred by Sputnik.
The first time I saw this movie was as a fifth-grader around 1984, when I was in a summer math class at Black Hawk College in Moline, Illinois. Since then, I’ve probably seen it 30 times.
As president of the undergraduate math club at The University of Chicago, we had an annual Donald viewing with Edwardo’s pizza. Then as a postdoc I found a VHS copy on eBay. Now I try to show it in every class.
The movie has some memorable scenes. The part on the golden ratio is quite interesting, showing first how it’s found in the pentagram, then how that ratio is found in nature. The second most memorable scene is the part about playing billiards “by the diamonds.” You can read the whole plot on Wikipedia if you like, although they seem to suck all the enjoyment out of in that writeup.
It’s quite dated–there are some high-tech items that are genuine artifacts by now, and Donald’s not exactly politically correct. But I’m glad it was made, and I hope it comes out on DVD sometime soon.
Youtube videos: part 1 part 2
technorati tags:math, funny, donald, duck, disney
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05.15.07
Posted in Math, Web, Pop Culture at 6:35 am by leingang
A while back I wrote that numb3rs is one of the most legitimate pop-culture portrayals of a mathematician. I’ve stopped watching the show, but some mathematicians still do, and Mark Bridger at Northeastern even blogs about it. So you can break down the math in each episode.
technorati tags:numb3rs, blog, math
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05.11.07
Posted in Pop Culture at 7:55 am by leingang
I am a child of the 80s (Atari, not Nintendo), so I’m very interested in the upcoming CGI Transformers movie. True, I’m married with two little ones, which means I go to the movies about once a year, twice if you count the annual Pixar product. Still, graphics are cool, and so I was pretty excited to see these shots of what the new transformers look like.
Here’s a shot from the old cartoon:
And now Optimus Prime and Bumblee today:
technorati tags:transformers, movie, cool, graphics
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Today marks the 50th anniversary of the launch of Sputnik 1, a mostly harmless satellite but only the first man-made object to orbit the earth. It had a profound effect not only on world politics but on education in the United States. As written in Air Force Magazine this month:
