Archive for the ‘Math in Pop Culture’ Category

Math in Bones

December 1, 2008

dvdSo I’ve been watching DVDs of Bones, as you might have gathered from an earlier post. And one of the episodes from Season 2, “Spaceman in the Crater”, has three separate spots with some math! Not a lot of math, but it doesn’t take much to make me happy.

The first bit comes early on, when Special Agent Booth and Forensic Anthropologist Dr. “Bones” Brennan examine a body that’s fallen from the sky and formed a crater.

Brennan (immediately after commenting on the man’s loafers): He hit the ground at approximately 200 kilometers per hour.
Booth: How can you tell that by his shoes?
Brennan: 124 miles per hour is terminal velocity for a falling human.

See that fancy unit conversion? It’s right, too. (OK, not hard math, but for all we know Brennan used Fibonacci numbers to figure it out, since 200=144+55+1 [i.e. F12 + F10 + F2] so the miles would be approximately 89+34+1=124 [i.e. F11 + F9 + F1].

Then a little later, Dr. Zack Addy took it a bit further:

A human being reaches terminal velocity after falling 200 to 220 meters, depending upon air resistance. Velocity would be achieved between 5 and 8 seconds, depending upon atmospheric conditions, body position, and clothing. He fell from a minimum of 1200 feet. I can run through the math if you like.

Sadly, no one wanted Addy to run through the math so I can’t see what played into this. Presumably if air resistance were negligible (which it isn’t) and if the body was dropped rather than pushed, then it would fall ½(9.8)t2 meters after t seconds; that’s 122.5 meters per second after 5 seconds and 313.5 meters per second after 8 seconds. The upper bound of 8 seconds makes sense to me, since air resistance would slow the speed down, but I’m a little surprised by the lower bound of 5.  Shouldn’t it be at least 6 seconds, and probably closer to 7?  Maybe Addy is thinking the body might have been pushed.

In the final math segment there’s a bit of exponential decay. Addy shares the following observation a bit later, after they’ve determined that the victim was an astronaut:

Astronauts lose 2% of their bone mass for each month spent in space. Our victim’s legs, hips, and lower vertebrae have demineralized over 20%, indicating 10 months in space.

Using that 2% per year, the amount of bone mass left should be (0.98)n after n months. Solving (0.98)n=0.8, in order to find out when the boned demineralized 20%, leads to n=log(.8)/log(.98) [where the log is to your favorite base], or n≈11.04 months. That doesn’t fit with the 10 months mentioned above. In this case, though, rounding could be the culprit: if it was really 2.4% of bone decay per month and that number was rounded down to 2% for convenience, the formula becomes (0.976)n=0.8, giving n=log(.8)/log(.976)≈9.2, which fits with the data. I’ll grant Addy this one.

(And yes, the rest of the episode was good too!)

Perfection

November 26, 2008

perfect-numbersIn the episode “The Boneless Bride in the River” in Season 2 of the TV show Bones, the body of a young woman is found in a river, and it’s discovered that she was likely brought over to the US on a fiancée visa. About 14 ½ minutes into the episode, two of the main characters (Special Agent Seeley Booth and Forensic Anthropologist Dr. Temperance “Bones” Brennan) have this conversation:

Booth: Homeland Security says the fiancée visa was expedited by a lawyer on retainer into a smaller bride agency here in town called “The Perfect Wife”.
Brennan: Oh that sounds archaic.
Booth:
No, you know, in therapy I learned that superlatives like perfect are meaningless.
Brennan:
Not in science. A perfect number is a number whose divisors add up to itself, as in one plus two plus three equals six.
Booth:
Well, in therapy I learned that definitive statements are by their very nature, wrong.
Brennan:
Isn’t the statement “definitive statements are by their very nature wrong”, definitive, and thus wrong?

Speaking of wrong, Brennan was a little bit wrong in her definition: a perfect number is one whose proper divisors add up to itself. But still, neat math in a neat show is always worth a mention. And perfect numbers are pretty neat, because like so much in number theory they’re simple but there are still open problems about them.

A bit of history [where "a bit" apparently means "a lot because I don't know how to edit today"]: perfect numbers were studied by Pythagoras, which makes the concept at least 2500 years old. Euclid also talked about perfect numbers a few hundred years later in Book IX of The Elements. In particular, Thomas Heath’s translation of Proposition 36 states:

If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect.

As a translation of this translation, this is saying that if 1+2+22+…+2k is prime, then that sum times the last number in the sum (2k) must be perfect. For example, 1+2=3 is prime so 3·2=6 is perfect. Likewise, 1+2+4=7 is prime, and 7·4=28 is perfect. [As a note, the formula is is sometime written out algebraically with 2k+1-1 used instead of 1+2+22+…+2k; in addition, when it’s written that way it’s sometimes reindexed so that k is used in place of k+1, and the statement becomes “If 2k-1 is prime, then (2k-1)·2k-1 is perfect.”]

A few hundred years after Euclid, Nicomachus wrote some more information about perfect numbers. According to the St. Andrew’s web site, he made
five claims:

(1) The nth perfect number has n digits.
(2) All perfect numbers are even.
(3) All perfect numbers end in 6 and 8 alternately.
(4) Euclid’s algorithm [described above] to generate perfect numbers will give all perfect numbers
(5) There are infinitely many perfect numbers.

Nicomachus’s word was law as far as perfect numbers were concerned, and his claims, while unproven, were believed for decades centuries a really really long time. Of course, there were only four perfect numbers known at that time (6, 28, 496, and 8128), so they really didn’t have much to go on. In reality, the 5th and 6th perfect numbers (33,550,336 and 8,589,869,056 respectively) disprove claims (1) and the “alternately” portion of (3), but it took a while for someone to discover those.

(Which leads to a little tangent: who did discover those numbers? Ismail ibn Ibrahim ibn Fallus (1194-1239) made a list of ten numbers he thought were perfect. Three of them weren’t, but the other seven were. Sadly, a lot of other mathematicians had no idea about this list: mathematicians in Western Europe had to wait another 350 years for those numbers to enter their collective psyche, during which time a few other mathematicians found the 5th and 6th perfect numbers and were equally ignored.)

But even after it was known that Nicomachus’s claims weren’t themselves perfect (ba DUM!), mathematicians continued to study the numbers. In the 1600s Pierre Fermat tried to find patterns, and ended up discovering his Little Theorem as a consequence. Marin Mersenne also spent some time on it, and in fact his exploration of when 2k-1 is prime, as a part of that theorem of Euclid’s mentioned above, led to the notion of Mersenne primes (primes of the form 2k-1 where k itself is prime).

In the 1700s Leonhard Euler entered the fray. He couldn’t prove that Euclid’s formula generated all perfect numbers, but he did show that it generated all even perfect numbers. And a bunch of other mathematicians spent a lot of time trying to show that numbers were or were not perfect (which was related to showing that specific numbers of the form 2k-1 were or were not prime), a challenging task in the pre-computer days.

Not that we’re doing much better now. As of this moment, we still only know of 46 perfect numbers, and they’re pretty big. We do know a few cool things about perfect numbers in general:

  • Even perfect numbers end in 6 or 8.
  • Even perfect numbers are triangular numbers (e.g. 6=1+2+3 and 28=1+2+3+4+5+6+7) where the ending digit is one less than a power of 2.
  • The reciprocals of the divisors of perfect numbers all add up to 2:
    \frac{1}{1} + \frac{1}{2}+\frac{1}{3} + \frac{1}{6}=2
    \frac{1}{1} + \frac{1}{2}+\frac{1}{4} + \frac{1}{7}+\frac{1}{14} + \frac{1}{28}=2

But there’s a lot we still don’t know:

  • We don’t know if all perfect numbers are even.
  • We don’t know if there are a finite or infinite number of perfect numbers.

In other words, perfection continues to eludes us.

Acting Tips

October 16, 2008

I was recently cleaning watching Friends, “The One with the Race Car Bed” from Season 3, and lo and behold there was some math! in this episode Joey got a job teaching “Acting for Soap Operas”, and in one scene he shares some tricks of the trade: how to cry, how to convey evilness, and how to respond to bad news. The math is in that last one: “Let’s say I’ve just gotten bad news. Well all I do there is try and divide 232 by 13.”

How well does that work?  [The math part starts about 40 seconds into the scene.]

Grey’s Anatomy Math

September 25, 2008

Season 5 of Grey’s Anatomy starts tonight, and the big question out there is, “Will there be any math on the show this year?” Because there was in Season 3. Remember how Izzie Stevens got a check for $8,700,000? Seven Episodes later, in Episode 11 (“Six Days — Part I”), Izzie still has the check on the front of her fridge, and the psychiatrist won’t clear her for surgery until she deposits it. But Izzie has gotten to know a patient who is going to have her spine straightened and she really wants to scrub in, so she confronts Miranda Bailey and tries to convince her. The scene ends with this :

Dr. Bailey: Did you deposit the check?
Izzie: It’s my money. I should get to do what I want with it.
Dr. Bailey: You get a 5% return on a 6-month CD?
Izzie: (says nothing)
Dr. Bailey: In the time we’ve been standing here you could’ve just made 400 dollars.

When Bailey made that last remark, they’d been standing there for 29 seconds. So what is the actual value of the interest Izzie would have earned during that conversation?

CDs generally seem to have interest compounded monthly, although if Izzie had only recently deposited the check the interest wouldn’t have had time to compound yet so we can ignore that. During the first month, assuming 5% annual interest, Izzie would earn $8,700,000(0.05/12)≈$36,250 in interest. If there were 30 days in the month, that amounts to $1208.33 per day (not “thousands of dollars a day in interest” as mentioned earlier in the episode — the interest rate would need to be about 8.3% for that kind of yield).

There are 86,400 seconds in a day, so that $1208.33/day translates to just about 1.4¢/minute, which only amounts to 40.6¢ in 29 seconds.

Dang, Bailey was way off. I guess, “In the time we’ve been standing here you could’ve just made 41 cents” doesn’t quite pack the same punch.

Line Rider Fun (or McDonald’s does Calculus)

September 8, 2008

Last year a few of our students discovered Line Rider — a computer game in which they could draw paths and a little sledder guy slides down and up. It’s a fantastic application of tangent lines (and therefore calculus with perhaps some vectors and physics), because in the version that I saw them doing last year they would simulate a curve by drawing lots of straight lines, as in this 83-second video:

It looks like Line Rider has gotten all fancy now (or maybe the artists are just more experienced) and you can draw curvy lines for the little guy to slide on. This inspired the following 30-second McDonald’s commercial:

Because calculus is so important, Line Rider has also been released for phones (Mobile in August and, according to a news release this past week, the iPhone and iPod Touch this month). No need to be without it ever!

Have you ever noticed how mathematicians count pretty much everything as applied mathematics? Yeah, I know, looking at the world through math-colored glasses!

More Jeopardy Math

September 5, 2008

I was recently reminded of my all-time favorite Final Jeopardy clue (I can’t remember the exact wording), which showed up many (>15) years ago:

This is the only positive number such that adding it to itself and multiplying it by itself give the same result.

The question, “What is 2?”, is a no-brainer for anyone who’s played with numbers, and I’ve played with numbers a lot, so I knew the question. What struck me, though, was that nobody got it right. Two people guessed 1, and one guessed 1.5. 1.5?! This is why we tell our students to check their work.

OK, so three Jeopardy contestants can’t do arithmetic in 30 seconds under bright lights with millions of people watching them. I can forgive that. (Maybe they’d have had better luck with this one.) But it got me wondering about how often math appears as a Jeopardy category. Luckily, there’s the J! Archive, which has just about every clue from every show ever aired. And it’s searchable! So here’s the list of all Jeopardy clues containing the string “math”. (Note that this includes the occasional clue with a word such as “aftermath” or “Mathew”. Also, my favorite clue isn’t in the archive – I said almost every clue.) Here are a few good ones:

  • (Mathematics, $1000): He never revealed the proof of his last theorem, but only hinted at it in a margin note. (answer)
  • (Math Terms, $600): It describes the shortest line between 2 points on a surface, or a dome made with the least possible material. (answer) (This is pretty sophisticated, don’t you think?)
  • (Milestones in Math, $600): This Persian poet’s work on algebra included systematic solutions to cubic equations. (answer, and his Rubaiyat)

Of course, once I’d found a searchable archive of Jeopardy clues, I immediately looked for my favorite category (no, it’s not Math): Stupid Answers. A few examples:

  • Oddly, this piece of furniture was never built for the throne room at Neuschwanstein Castle (the throne)
  • In the musical “The Phantom of the Opera”, the Phantom writes one of these, called “Don Juan Triumphant”. (an opera)
  • He formed United Artists with Griffith, Pickford & Fairbanks & in 1915 lost a Charlie Chaplin look-alike contest. (Charlie Chaplin)

I love Jeopardy. If nothing else, it serves as a reminder of just how much I don’t know. But I enjoy when they decide to include math clues, because at least I have a chance.

Six Degrees of Separation?

August 8, 2008
Photo of Kevin Bacon by SAGIindie (Creative Commons license)

Photo of Kevin Bacon by SAGIindie (Creative Commons license)

The notion of “degrees of separation” is once again in the news, thanks to Microsoft. The basic idea is that you start with someone, look at everyone they know (where “know” would have to be defined in some way), then look at everyone that those people know, and see what happens. Chain letters and pyramid schemes thrive on this sort of thing. The origin of “six degrees of separation” seems to date back to the 1929 short story “Chains” by Frigyes Karinthyin:

One of us suggested performing the following experiment to prove that the population of the Earth is closer together now than they have ever been before. We should select any person from the 1.5 billion inhabitants of the Earth—anyone, anywhere at all. He bet us that, using no more than five individuals, one of whom is a personal acquaintance, he could contact the selected individual using nothing except the network of personal acquaintances.

Musing #1: The 5th root of 1.5 billion is about 68.5. This suggests that each person would need at least 68 personal acquaintances if there was no overlap. But since many people share personal acquaintances, the actual number would have to be a lot higher.

Musing #2: The notion of six degrees of separation was introduced in a work of fiction.

Then there was this paper by Jeffrey Travers and Stanley Milgram in 1969 about an experiment by Milgram in which people in the USA got letters to someone they didn’t know by sending the letters to people they did know, who would send it to people they knew, etc. The average number of people it passed through was 5.2 [so each letter was sent 6.2 times on average to get to the final person] leading to more emphasis on that whole SIX thing.

Problem #1: Here the 6.2 was an average, not a maximum like in the short story above. This distinction is always sometimes pretty much always blurred when degrees of separation are talked about informally.

Problem #2: The 6.2 was the average only of the letters that made it to their destination, and that was only 64 out of 296. In other words, this experiment suggests that two people in the US would be 6.2 steps on average away from each other if they’re connected at all, but they probably aren’t.

Problem #3: The title “Infinite Degrees of Separation” isn’t so catchy, and is a little depressing.

Problem #4: This is the same Stanley Milgram who conducted the infamous Milgram Experiment in which volunteers gave supposed electric shocks to supposed volunteers in response to mistakes. Not that this has anything to do with the Small World theory, but it was a surprise to me that it was the same guy.

Moving along, the degrees of separation results were repeated in the Small World Project through Columbia University and published here in 2003. As explained in the abstract:

We report on a global social-search experiment in which more than 60,000 e-mail users attempted to reach one of 18 target persons in 13 countries by forwarding messages to acquaintances.

This was worldwide, not just the USA, and with much bigger numbers, and yours truly actually participated in it. It led to some interesting results:

Result #1: Of the completed chains, the median number of steps was 4.05, though the authors conjectured that overall the median number of steps is probably five to seven. This lead to headlines like “Email experiment confirms six degrees of separation“.

Result #2: The participants were not a random sample. From the article above: “More than half of all participants resided in North America and were middle class, professional, college educated, and Christian” In theory, I think only the starting and ending points would have to be random, but here the starting and ending people did need to have internet access and I’d question whether or not internet access is truly independent of social connectedness internationally.

Result #3: The average is, once again, based only on completed chains. And out of the 24,163 chains, only 384 were finished. In other words, most people don’t seem to be connected. Either that or we’re just lazy — there’s some evidence for that in the paper: if people couldn’t think of a good person to pass the email along to, they didn’t pass it along to anyone.

Then in June 2006 Microsoft got in on the action. They looked at all the Instant Messenges that month and counted folk as connected if they exchanged at least one IM. The average degrees of separation was 6.6, which lead to recent headlines like, “Everyone really is just six degrees of separation from Kevin Bacon” with the subheader, “A study of Microsoft’s instant messaging network supports the popular idea that everyone on the planet can be connected through fewer than seven links in a chain of contacts.”

Misconception #1: Once again, the six degrees, which was really more like seven, is an average and not a maximum. Microsoft did find that 78% of users could be connected in seven stages or less, but they also said that some pairs were separated by 29 degrees. Actually, what might be the most interesting bit to me is that the news articles I looked at (on Yahoo! News and The Washington Post) imply that everyone was connected in some way, which is very different from the results of the previous study.

Misconception #2: I think it’s a stretch to claim that Microsoft IM users are representative of everyone on the planet, and I’d be wary of claims like the headline above that extrapolate it to every single person.

But despite the limitations of studies like this, if you limit yourself to certain subsets of people, you still can have some fun. For example:

Related Website #1: The Oracle of Bacon. This site allows you to choose an actor and see that actor’s “Bacon Number” (their degrees of separation from Kevin Bacon). For example, if you pick Christina Mastin, you find:

Christina Mastin has a Bacon number of 2.

Christina Mastin was in Dogfight (1991) with Brendan Fraser
Brendan Fraser was in Air I Breathe, The (2007) with Kevin Bacon

You can also choose the Advanced Setup to search for a path linking any two actors. Incidentally, the notion that all actors are connected to Kevin Bacon appeared around 1994, and last year Kevin Bacon started a charitable giving organization called SixDegrees.org, which is a nice thing even though Kevin himself claims on the site that the six degrees of separation is a maximum and not an average. (You might want to correct that, Kev. But still, kudos for the charity work.)

Related Website #2: The Erdös Number Project. This site studies collaboration among mathematicians, many of whom have co-authored a paper with someone who co-authored a paper with someone who co-authored a paper with Paul Erdös. For example, Natalie Portman has an Erdös number of 5: she published a psychology paper (as Natalie Hershlag) with Abigail A. Baird, who published a paper on Functional Connectivity with M.S. Gazzaniga, who published a paper on Acquired central dyschromatopsia with J.D. Victor, who published a combinatorics paper with J. Gillis, who published a paper on transfinite diameter with Paul Erdös. Incidentally, Natalie currently has a Bacon number of 2 but this coming February she and Kevin are both appearing in the movie New York, I Love You, giving Natalie an Erdös-Bacon Number of 6 (5+1).

Related Website #3: The six degrees of celeb dating. This site is a game, so it gives you two people and you have to fill in the dating chain yourself. The site will, however, tell you if you’re right and it gives you a limited number of celebrities to choose from: I was able to tell that Drew Barrymore dated Edward Norton who dated Salma Hayek in only about ten guesses. If that’s too much work or if closed cycles interest you more than chains, you can see a twelve-step dating cycle starting and ending with Brad Pitt on cbs2chicago.

Order of operations: does it really matter?

July 31, 2008
George Orwell

George Orwell

While surfing the webpages of a variety of newspapers this morning, I stumbled on the following….

On the staged-reality-tv show Big Brother (UK version), they gave the housemates a mathematical task: they had to compute three different sums, then use the three resulting answers as the combination to a safe.

Implicit in the problem was that the three calculations should each result in a two-digit integer.

Video posted to the Channel 4 website shows the contestants muttering, struggling, and having an extremely difficult time of it. And with good reason!

Here are the three calculations they were given, as posted on the Channel 4 website:

Sum#1: 3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

Sum#2: 1396 x 2 ÷ 4 — (12²) + 46 x 2 ÷ 40 x (5²) – (7 x 99) x 3 – (11²) x 5 – 219

Sum#3: 100 – 33 x 5 + 665 ÷ (5²) x 17 – 248 x 3 ÷ (4²) + 52 ÷ 7 + (273 – 217)

From the look of things on the on-line video, I’m guessing that the contestants had no writing implements, and had to do all of this in their head. That makes this challenging enough, I suppose.

Making matters worse is that none of the three sums is integer valued; they work out to 62/5, -4583/2, and 26189/70   28289/70. Rather, these are their values if one computes using the usual order of operations, where exponents have precedence over other operations, where multiplication and division take precedence over addition and subtraction, where calculations are performed left-to-right, and parentheses can be used to override this sequencing. (“PEMDAS” is a popular acronym with my students, standing for “Parentheses, Exponents, Multiplication and Division, Addition and Subtraction”, and sometimes recalled using the mnemonic “Please Excuse My Dear Aunt Sally”)

Apparently the folk who created this puzzle expected their contestants to work left-to-right, ignoring operator precedence, in the way that a $1 calculator might do. (Calculators that do pay heed to order of operation conventions are often marketed as “scientific” calculators.)

For example, the first sum should go as follows:

3 x 17 – 24 + 78 x 9 ÷ 5 – (13²) + (65 – 29) ÷ 4 + (4²) – (7 x 3) + (3²) + 99 – (7²) – 49

= 3 x 17 – 24 + 78 x 9 ÷ 5 – 169 + 36÷ 4 + 16 – 21 + 9 + 99 – 49 – 49

= 51 – 24 + 702/5 – 169 + 9 + 16 – 21 + 9 + 99 – 49 – 49

= 62/5

But I suspect the intended calculation was instead:

3×17 = 51, 51-24 = 27, 27+78 = 105, 105×9 = 945

945÷5=189, 189-(13²)=20, 20+(65-29)=56. 56÷4=14,

14+(4²)=30, 30-(7 x 3)=9, 9+(3²) + 99 – (7²) – 49 = 19

Similar (incorrect!) computations for sum #2 and sum #3 yield 31 and 75, respectively.

Clearly it is important that we agree on our order of operations.  But why do we prefer one over the other? Is this merely a cultural convention?  One stock answer to this is to point to the algebra of polynomials: our conventions regarding operator precedence play a central role in how we interpret linear equations (e.g. what is the slope of the line y=3+4x?), how we interpret polynomials (e.g. is 4 - 3x + 7x^2 a quadratic or a cubic polynomial?), and how we compute sums and products of polynomials.

But this morning, having not yet had my first cup of coffee, I wonder: is it possible to change the rules of arithmetic, so that all operations have the same precedence (unless exceptions are forced by parentheses), and to develop a meaningful algebra based on similar principles? It seems to me that the answer is yes, and I wonder exactly what is lost by doing so, other than familiarity.

Calculus in the tannery

March 21, 2008

cowskin.jpgHow do you calculate the surface area of an irregular figure? If it’s a cowskin, you can buy an Electronic Area Measuring Machine to find it for you! The LOTO/SOFT model, for example, features continuous measure modeling, only 20 millimeters between photoelectric elements, and output in m², dm², tenths of dm², quarters of a square foot, tenths of square foot, or square inches. Not the right model for you? Don’t worry, there are several more to choose from.

I’m not completely sure how these machines work, but I’m guessing from the pictures and descriptions that each photoelectric element measures the length of skin that passes under it. Then, using the fact that the sensors are 20mm apart (so Δx=20mm) the machine must use the Trapezoid Rule, Simpson’s Rule, or some other form of numeric integration to estimate the surface area. Hooray — calculus in action!

Thanks to Mary Louise for telling me about this last week. She saw it on the “Tannery” episode of Dirty Jobs.

“Johnny”, or the interaction of odds and probability

March 15, 2008

I was recently listening to the album Johnny the Fox, recorded by the second most famous Irish rock band. The opening song on the album, “Johnny”, is a tale of a young man being pursued after having committed a crime. The climax of the song finds him in an alley, surrounded by police, and the song’s narrator describes his potential fate:

Five to one he gets away

That’s the odds I’m gonna give

Five to four they blast him away

Three to one he’s gonna live

But what exactly does that mean? Giving odds of 3 to 1 means that if someone were to bet that Johnny gets away, for each 1 dollar that was wagered, the house would pay winnings of 3 dollars. If I offer 3:1 odds, I must believe that for every 1 dollar winning bet, there will be a matching 3 dollars bet on losing outcomes. Thus 3 to 1 odds corresponds to the judgement that 1 out of every 4 dollars wagered will be on that particular outcome, a probability of 1/4.

In the case of the song “Johnny”, we find that the odds being offered (5:1, 5:4, and 3:1) correspond to the probabilities 1/6, 4/9, and 1/4, which add to (6+16+9)/36 = 31/36. Since these probabilities don’t add to 1, this isn’t a zero-sum game, and someone has the advantage. But who?

When it comes time to pay off the bets, the situation is quite clear. One of the potential outcomes has actually occurred, and the payoff odds (assuming a zero-sum game) will simply be the ratio of the total money wagered divided by the amount wagered on that outcome.

In the case of three possible outcomes, if we assume that the total wagers have been $a, $b, and $c on the three outcomes, and set $s = $a+$b+$c, we’d have odds of

s-a:a

s-b:b

s-c:c

with corresponding probabilities a/s, b/s, c/s, adding to 1.

Alternately, if the house takes a cut of the total wagers as profit (as happens at parimutuel betting facilities, such as race tracks, in the USA), the amount of money available to pay to gamblers isn’t $s, but rather $s-p, where $p is the profit from the race for the house. In that case, the odds offered would be

s-p-a:a

s-p-b:b

s-p-c:a

whose probabilities would be a/(s-p), b/(s-p), and c/(s-p), with sum s/(s-p), a probability that is slightly more than 100%.

In general, we’d expect the sum of the probabilities from the odds to add to more than 1, and it appears that the scenario described by lyricist Phil Lynott in “Johnny” isn’t one he will profit from. Good thing Phil was a musician and not a bookie.

In the first race on March 14 at Bay Meadows Race Track, the actual odds after the race were 21:10, 309:10, 3:5, 31:5, and 9:1; as probabilities, these become 10/31, 10/319, 5/8, 5/36, and 1/10, whose sum is 4335479 / 3560040 = 4335479 / (4335479 – 775439), so in practice they had set aside $755439 out of every $4,335,479 wagered (roughly 18%).

There are several notions of probability lurking here. The bookie setting odds is trying to anticipate how gamblers are going to wager. In theory, gamblers are trying to anticipate the actual likelihood of each outcome occuring, then comparing that to the odds offered, and maximizing their expected value. But the bookies know this, and act accordingly. And from there it gets complex.

Shades of: In theory, theory and practice are the same. In practice, they are quite different. (Paraphrasing Einstein, or Popper, or Yogi Berra, or any one of a number of 20th C philosophers….)

A “Paltry Geometric Dilemma”

February 23, 2008

One of the greatest “math” scenes in a movie appears in Better Off Dead, starring John Cusack. Watch his geometry teacher (Vincent Schiavelli) in action (transcript below).

The three cardinal trapezoidal formations hereto made orientable in our diagram by connecting the various points HIGK, PEGQ, and LMNO, creating our geometric configurations, which have no properties, but with location (Ohh!) are equal to the described triangle CAB quintuplicated. Therefore, it is also the five triangles composing the aforementioned NIGH – each are equal to the triangle CAB in this geometric concept! (laughter)

Therefore, in a like manner the geometric metaphors can derive a repeated vectoral sum. This was your assignment, and I would like to see the results.

My favorite line: “which have no properties.” Indeed.

Those Boring Venn Diagrams

February 19, 2008

Eddie Izzard on Venn and his diagrams. The first 45 seconds are the mathy bit. (Warning: NSFW)

“The Comedy of Science”

February 12, 2008

Go watch this clip from Robot Chicken. It’s only 10 seconds long, but it’s really funny. Does anyone want to volunteer to figure out if what they’ve written on the board is legitimate? (Warning: The clip is safe for all audiences, but no guarantees are made for the rest of the site.)

A Matter of Perspective

January 24, 2008

Sunday’s F Minus (click for full size):

F Minus perspective

The “Law of Probabilities”

January 17, 2008

This Pearls Before Swine strip appeared on December 29, 2007 (click for full size):

For a related discussion, see the Infinite Monkey Theorem.


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