[hrothgar]

FrancesHinden, on Sep 23 2005, 05:23 PM, said:

As an alternative, I agree that the idea of balancing movements, or how many rounds of Swiss do you need to get to the winner, or what's the best format for the Bermuda Bowl... are all mathematically interesting problems that are easy to explain. They aren't really related to _playing_ bridge, however.

This is a great suggestion:

I find the relationship between statistical sampling and tournament design fascinating.

If you decide to go this way, I suggest reading a paper called "Selection in the Presence of Noise: The Design of Playoff Systems" by Adler, Gemmel, Harchol, Karp, and Kenyon.

[cherdano]

FrancesHinden, on Sep 23 2005, 04:23 PM, said:

Like everyone else, I suggest Restricted choice. It doesn't take long to explain the problem.

If they know Bayes' theorem, you can show how this is just a simple example: it's just the terminology that's different "he's more likely to have singleton Q than QJ because with the latter he would have had a choice of cards to play" is not obviously the same thing as the standard algebraic statement of Bayes' theorem using set notation.

And if they don't know Bayes' theorem (which I would rather assume), it is an excellent opportunity to learn one incarnation of it.

If I had to pick a single mathematical statement that should be known more widely, there would be no second choice to Bayes' theorem. (E.g. I think it should be used more often for hypothesis testing, since all these 95% confidence tests don't do what people think they do.)
And bridge players are the only group outside of mathematics that I know who have, in a way, formed their own understanding of it (under the name of PRC).

Arend

[Al_U_Card]

Possibly you might consider the use of simple hand frequency as a determining factor in bidding methods. The choice of one bid over another should depend on that hand occuring the most often? (This is not always the case, I would imagine.) So, in real life, straight calculations do not always result in a highly correlated response......

[kfgauss]

There's a great article on mixed strategies in cardplay by John Swanson at www.johnninaswanson.com/articles.html called "A Hand for Deep Blue."

He analyses the position Axx opposite KJ9xx and determines that a mixed strategy is required for the defenders regarding when they should falsecard the 10. It's a short article, so I won't summarize -- just go read it.

The position has a nice history too (see the notes below the article), with a writeup by Edgar Kaplan in the Bridge World after it came up in the Vanderbilt and led to an accusation of cheating (Kaplan's claim was that the accused made the right play, actually).

Andy

[tysen2k]

I agree that something along the lines of restricted choice is best. Since almost no one will have bridge experience don't even touch bidding or hand evaluation.

Tysen

[FrancesHinden]

cherdano, on Sep 26 2005, 10:29 AM, said:

And if they don't know Bayes' theorem (which I would rather assume),

Rant on: I learnt Bayes theorem at school. I've no idea if it's still part of the (UK) A-level further maths syllabus or not, I imagine it isn't. And yet they tell us standards continue to rise....

sorry, have to say things like that sometimes....

[hrothgar]

FrancesHinden, on Sep 26 2005, 08:55 PM, said:

cherdano, on Sep 26 2005, 10:29 AM, said:

And if they don't know Bayes' theorem (which I would rather assume),

Rant on: I learnt Bayes theorem at school. I've no idea if it's still part of the (UK) A-level further maths syllabus or not, I imagine it isn't. And yet they tell us standards continue to rise....

sorry, have to say things like that sometimes....

Rant seconded...

I'd be shocked if the target audience (Juniors and Senior level math majors at Michigan) hadn't seen Bayes theorem. If they haven't, then something is seriously wrong in the state of Michigan...

[kfgauss]

hrothgar, on Sep 26 2005, 06:08 PM, said:

I'd be shocked if the target audience (Juniors and Senior level math majors at Michigan) hadn't seen Bayes theorem.  If they haven't, then something is seriously wrong in the state of Michigan...

Statistics really isn't part of pure math programs these days. It would usually only be required coursework if you're getting a degree in "applied math" (or a math degree with an "applied" focus) or if the department doesn't make any distinction between pure and applied math. This means that most of the students graduating (in pure math) from the top math schools have usually never taken a statistics course (unless they took one in high school, which is reasonably common but would be very basic). That said, most of them will understand Bayes Theorem anyways (informally, perhaps). Feel free to feel about this as you will.

Andy

[han]

hrothgar, on Sep 26 2005, 01:08 PM, said:

FrancesHinden, on Sep 26 2005, 08:55 PM, said:

cherdano, on Sep 26 2005, 10:29 AM, said:

And if they don't know Bayes' theorem (which I would rather assume),

Rant on: I learnt Bayes theorem at school. I've no idea if it's still part of the (UK) A-level further maths syllabus or not, I imagine it isn't. And yet they tell us standards continue to rise....

sorry, have to say things like that sometimes....

Rant seconded...

I'd be shocked if the target audience (Juniors and Senior level math majors at Michigan) hadn't seen Bayes theorem. If they haven't, then something is seriously wrong in the state of Michigan...

Richard, you haven't looked at my profile lately, I am a badger now, I'm no longer in Michigan.

BTW, I would be schocked if any given reasonably large group of students all know a single fact. I can't imagine that they all know Bayes theorem. However, that makes it even more important to mention it.

For those who don't remember names of theorems they learned in high school as well as Frances:

Bayes theorem says that the chance for A to be true given that B is true is equal to the chance that both A and B are both true divided by the chance that B is true. In formulas:

P(QJ doubleton offside| Q or J appears at trick 1) = P(QJ doubleton)/P(QJ doubleton or Q singleton or J singleton) ~ 1/3.

[pigpenz]

have them read Jim Gliecks book "the theory of chaos"

a coastline is infinite! the further you go into it, same with bridge some things look simple but the more you look at it they really are very complex.

[matmat]

heh... you could always start with the basics -- the mathematics of shuffling. perfect shuffless? what is a fully randomized deck? etc.

[FrancesHinden]

kfgauss, on Sep 26 2005, 02:01 PM, said:

Statistics really isn't part of pure math programs these days. It would usually only be required coursework if you're getting a degree in "applied math" (or a math degree with an "applied" focus) or if the department doesn't make any distinction between pure and applied math. This means that most of the students graduating (in pure math) from the top math schools have usually never taken a statistics course (unless they took one in high school, which is reasonably common but would be very basic). That said, most of them will understand Bayes Theorem anyways (informally, perhaps). Feel free to feel about this as you will.

The only comment I'd make is that Bayes' theorem is not statistics, it is probability, which is a branch of pure mathematics.

This is way off subject, but it's interesting to note that typically English Universities are considered to specialise far earlier than American ones. I read maths in England (at a place known on rgb as "Frances's University"), and in the first year it was compulsory to do the full range: analysis, algebra, probability, mechanics, statistics, quantum mechanics & special relativity (I've probability forgotten your favourite subject, but that doesn't mean it wasn't covered). You couldn't specialise at all until the second year.

Even more off topic: in my second year, I was lectured in relativistic electrodynamics by one of the few chess GMs who is a serious bridge player.

[han]

My math-bridge talk is this afternoon. This is my plan:

Of course, I will start by mentioning that the world championships started yesterday, and that the two US teams are playing eachother right now (I wish I could watch ). I will tell them where to go for free online broadcasts (swangames of course! ).

I will then say that studying math won't help you at bridge, nor does it work the other way around. However, I think that math and bridge experts share some characteristics that I might touch upon (most noticably imo is the fact that they take their subject very seriously, while most of the world will think that they are nuts).

I won't explain all the rules of bridge, and in particular won't talk about bidding unless it turns out that a large part of my audience already knows bridge. Then I might talk about relay systems and fibonacci numbers at the end of my lecture.

What I will explain is how the hands are played, with an open dummy. I will restrict to one suited bridge, i.e. suit combinations. I will start with this fascinating suit combination: AKQ2 vs 543. First we estimate the chance that the suit splits: 6 choose 3 divided by 2^6. Then I explain how the fact that bridge hands have exactly 13 cards implies that this is actually an underestimate, and I will calculate the exact odds (yes, by hand).

I move on to a simple finesse, AQ opposite xx.

If this all goes well I will explain restricted choice, using the example combination that Gerben (I think) suggested: AK432 vs 10987. I will show how this is related to the Montey Hall problem and how Bayes theorem can be used. Notice that so far the only field of mathematics that I have used is probability theory. I can explain all this to any beginning bridge player in 5 minute, but I expect that it will take a long time.

I might now hint at how game theory appears in bridge by showing that the person with QJ doubleton should use a mixed strategy: If you always play the same card then declarer's odds for getting it right increase slightly.

I might finish by showing a full hand: ♠xxx ♥AQxx ♦AKQx ♣xx opposite ♠ AKQx ♥xxx ♦xxx ♣Ax. The heart suit offers better chances than either the spade suit or the diamond suit, but the chances can actually be combined to get much better odds.

OK, now all I have to do is print out this page and I will have my lecture notes.

[han]

To continue Frances' rant:

I got my undergraduate degree in the Netherlands only 5 years ago, and at that time the first year curriculum was exactly like Frances described: we learned a little bit of every subject in the first year, including probability theory. Unfortunately the high school program continues to improve, so now the first year math students first need to be taught more basics. As a consequence complex analysis has been completely thrown out of the basic program (meaning, there are actually people in Holland who graduate in mathematics and never have studied complex analysis, a scary idea!).

I promise not to make more math comments that have nothing to do with bridge. At least not for a while.

[Al_U_Card]

Hannie, on Oct 24 2005, 02:33 PM, said:

there are actually people in Holland who graduate in mathematics and never have studied complex analysis, a scary idea!).

I got my undergraduate degree in Chemistry 30 years ago and I can't even remember basic chemistry!...Is this an age-old problem or an old-age one? btw good luck with your lecture and remember to use humor to keep down the snoring noises...lol

[han]

Sorry Peter, there must be a misunderstanding, I never joke.

[Al_U_Card]

Hannie, on Oct 24 2005, 03:17 PM, said:

Sorry Peter, there must be a misunderstanding, I never joke.

Then you better have your ear-plugs ready........

[han]

Well, my lecture was about an hour and I quit after discussing restricted choice. Amazing how long it takes to explain a finesse to people who are not used to playing card games. It was fun though.

[mycroft]

Too late, but a cute one - why is 4432 more likely than 4333? Simple combinatorics, I know, but it's a good real-life example that "seems odd".

If enough of the audience have played trump games, the Mississippi Heart Hand is good for a instruction on statistical evaluation (usually more is better, but there are degenerate cases).

Signalling - especially "how to read a six".

If it's a comp. sci-heavy audience, investigate the "best way" to compute scores, and the "best way" for directors to input scores (including what checking can be done vs. time for input). It's amazing how many students, when given a data structure problem (sort, compute, find) will do what the course says is "theoretically best" or "worst-case best", totally forgetting that Order(n) calculations are deliberately set in the limit as n grows to infinity. Here, of course, a simple lookup table plus overtrick calculation will be best - perhaps even just a simple (if larger) lookup table.

I like the idea of showing "combining chances"...

Michael.

[Deanrover]

- Ranking systems
- EV of partnerships (in terms of IMP/board) and variance.