I have left Harvard as of July 1, 2008 to take a position at NYU. This website has been cached and left static. Feel free to browse my new website, aka "What the heck is a Clinical Associate Professor?"

07.05.07

Food + Math = Fun

Posted in Math E-304 at 2:28 pm by leingang

There’s a nice Math Problem on MetaFilter right now, inspired by an Arby’s promotion. If you can select five items from a set of eight, with repetitions allowed, for one price, how many possible meals can you eat? (The answer is 792).

It reminds me of a problem I put on the Math E-304 final this year:

Harvard Square has three restaurants that sell burritos: Qdoba, Felipé’s, and Boloco. I’ve decided to eat a burrito for lunch every day of the week. How many ways can I do that if:

  1. I have no preference how often I visit each restaurant?
  2. I want to eat at Felipé’s three times and Qdoba twice?
  3. I want to eat at each no more than twice?

Answers after the jump, if you want to try them.
(HT: 43 Folders)

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05.23.07

Lottery Mania

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:

LOTTERY RESULTS
TEAM
REC.
ODDS
32-50
5.3%
31-51
8.8%
30-52
11.9%
22-60
25.0%
24-58
19.9%
28-54
15.6%
32-50
5.3%
33-49
1.9%
33-49
1.9%
33-49
1.8%
35-47
0.8%
35-47
0.7%
39-43
0.6%
40-42
0.5%

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 p_i 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 \frac{p_i}{1-p_j}.  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

f(i,j,k) = p_i \cdot \frac{p_j}{1-p_i} \cdot \frac{p_k}{1-p_i-p_j}\,\,.

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

\sum_{i,j,k} f(i,j,k)\,\,.

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.)

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05.11.07

Upcoming: MSRI conference on teaching teachers

Posted in Teacher Training, Education, Math E-304 at 8:13 am by leingang

I’m really looking forward to the Critical Issues in Education: Teaching Teachers Mathematics conference at the Mathematical Sciences Research Institute coming up at the end of the month.

This workshop will focus concretely on courses, programs and materials that aim to increase teachers’ mathematical knowledge for teaching. Both courses and programs that lead to initial certification and professional development of current teachers will be examined at the workshop. In addition, the workshop will examine efforts by colleges, universities, school districts, professional organizations and funding agencies to support people who teach these courses or lead these workshops.

I work with the Harvard Extension School’s Master of Liberal Arts in Mathematics for Teaching program, which aims to acquaint schoolteachers with higher mathematics. The goal is not so much to give them material that will be directly applicable to the classroom, but to stretch their minds mathematically so they will be able to stretch their students’ minds.

This past semester I’ve taught a probability course in this program using the Moore method. It’s been a real challenge for my students and me because I’ve never done that before. But I found it enjoyable and the students seem to have received it well, too. My colleague Bret Benesh and I will be presenting at this conference on our experiences with Inquiry-based courses in the ALM program.

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