Chance News 13

Do not put your faith in what statistics say until you have carefully considered what they do not say.

William W. Watt

Forsooth

Great howler for Chance News!

Steven Strogatz suggested the following:

Events like this send a shiver down the spine, but the math behind strange coincidences shows that most people simply have a poor grasp of statistics. The odds against meeting someone else at a party with your birthday are not 365 to 1. In a room with just 23 people, the chances that two of them will share the same birthday are better than even.

William Grimes reviewing Beyond Concidence

by Martin Plimmer and Brian King

New York Times, 20 Jan. 2006

Alcohol is now 49% more affordable

Here is a Forsooth from the January 2006 issue of RSS News with a comment by the editor.

Alcohol is now 49% more affordable than it was in 1978

Sky News
20 November 2005

[readers are invited to send in their
suggestions as to the exact definitions
of a and b in the equation b = 1.49a]

Note from JLS.

The concept of an affordabiliity index is used in many fields but it is often difficulty to find a definition. Here are a couple of examples:

In the Ottawa Citizen Dec 23, 2005 we read:

The RBC (Royal Bank of Canada) housing affordability index, which measures the proportion of pre-tax household income needed to cover the costs of owning a home, was 24.6 per cent for a condominium, which remains the most affordable type of housing; 28.8 for a standard townhouse, 35.5 for a standard bungalow, and 41.3 for a standard two-storey home.

And in the US congress College access and opportunity act of 2005 we read:

The college affordability index shall be equal to

(A) the percentage increase in the tuition and fees charged for a first-time, full-time, full-year undergraduate student between the first of the 3 most recent preceding academic years and the last of those 3 academic years; divided by

(B) the percentage increase in the Consumer Price Index-All Urban Consumers (Current Series) from July of the first of those 3 academic years to July of the last of those 3 academic years.

Do Superfluous Medical Studies Exist?

In Chance News 12 can be found an item Superfluous Medical Studies, It references David Brown's articlein the Washington Post of January 2, 2006 in which he "looks at several instances where...the evidence is so convincing that no more studies need or should be done." His phrase is "What part of 'yes' don't doctors understand." In particular, "he cites the use of aprotinin in heart surgery" which since 1987 had 64 studies each conclusively showing that aprotinin reduced bleeding. Researchers were criticized for persisting in evaluating aprotinin without being fully aware of the previous research.

Less than four weeks later on January 26, 2006, the New York Times and the Wall Street Journal had respective headlines, Doctors Urge Ending Use of Heart Surgery Drug and "Serious Risks Are Found In Heart Drug." The heart drug, Trasylol, is, as you might guess, aprotinin! Some 4374 patients were in the study published by the New England Journal of Medicine--"1295 were given aprotinin and 1705 one of two other drugs, both generics [of older drugs]" There was also "A control group, 1374 patients" who "had no drugs to prevent bleeding."

According to the Wall Street Journal, "About 29% of the Trasylol patients suffered a stroke or heart-related complication, compared with about 21% of the patients taking the generic drugs and 19% getting no drugs." The breakdown comparison between Trasylol and the alternatives is as follows:

5% of patients on Trasylol required kidney dialysis vs. 1% of those on one of the two alternative drugs.

16% of Trasylol patients had a myocardial infarction (heart attack) vs. 12% and 13% of those on alternatives.

9% of Trasylol patients experienced heart failure vs. 6% and 5% on alternatives.

Furthermore, "The researchers also found that cheaper alternatives ("$10 to $50" per patient) to Trasylol ($1000 per patient) were just as effective in limiting blood loss." The New York Times wrote, "The [New England Journal of Medicine] article said that halting aprotinin use globally would prevent 10,000 to 11,000 cases of kidney failure a year and save more than $1 billion a year in dialysis costs as well as nearly $250 million spent on the drug itself." In addition, "The study is significant because it was conducted without drug-industry funding at 69 medical centers, including many of the top U.S. hospitals."

Naturally, Bayer, the manufacturer of Trasylol, a drug which "in the first nine months [of 2005]" had a "world-wide sales of just less than $200 million" and is used in 150,000 patients in the U.S., "believes that Trasylol is a safe and effective treatment." Though the study was large, Bayer points out that it was observational and not a randomized trial. According to the New York Times, the implication is, "Doctors might have assigned sicker patients to a particular drug that could make the results for that drug look bad." However, the Wall Street Journal claims that a randomized trial could not be done because "many hospitals have deemed Trasylol effective and would consider it unethical to allow a patient to get a placebo instead." The study thus resorted to " 'match' like patients across the study groups to control for other risk factors, such as age, gender and additional health problems."

About the only unambiguous conclusion that can be reached is to take good care of your heart so that you avoid the need for by-pass surgery and the associated medication.

Submitted by Paul Alper.

Using bill-tracking data to predict epidemics

Web game provides breakthrough in predicting spread of epidemics, BJS, Science Blog, 6-Jan-06.
Money-tracking web-game informs mathematical model of epidemics, BoingBoing, 26-Jan-06.
Statisticians Count Euros and Find More Than Money, Otto Pohl, New York Times, 2 Jul 2002.

The study of how epidemics spread worldwide is critical for controlling diseases, particularly pandemics like AIDS, bubonic plague, SARS and avian flu. Scientists hope to improve their models for epidemics by unravelling the statistical laws of human travel in the United States and elsewhere using novel data gathering techniques that exploit the ubiquity of the internet.

For the US source, the data is collected at the www.wheresgeorge.com bill-tracking website. This is a web-game that encourages people to track the serial numbers of dollar bills as they move around the US. Where's George? players mark their bills with WHERESGEORGE.COM then visitors to the site are encouraged to enter the serial number of the bill they've found and where they got it. In this way, the passage of a dollar-bill (or some other piece of infection) can be tracked around the country. One of the authors of the Nature article (Brockmann) says:

We recognized that the enormous amount of data, as well as the geographical and temporal resolution of bill-tracking, allowed us to draw conclusions about the statistical characteristics of human travel, independent of which means of transportation people use.

Movements of money is similar to phenomena like "on hold" times at call centers and stock price movements. These are systems whose development depends largely or entirely on the previous state the system was in. But how does one model this mingling? Until recently, models of the geographic spread of disease were based on the assumption that viruses disperse over geographic areas in a way similar to the diffusion of fine dust particles on the surface of water or gas molecules mixing in a room. Competing models include using a complex set of equations to describe interactions between people that depend on distance between them, the number of people involved and physical or economic constraints.

Using the US game data, the Science blog claim that scientists have developed a mathematical theory that describes the observed movements of travelers amazingly well over distances from just a few kilometers to a few thousand. By analysing the data from the bill-tracking website, they found that money follows what are known as universal scaling laws (from local to regional to long-distance scales). Like money, viruses are transported by people from place to place. Because the mechanisms of transmission of diseases from human to human are already well understood, the scientists can use these novel data sources to develop better models to better explain the global spread of a disease, during an epidemic. Another of the authors (Hufnagel) explained

Since we can't track people with tracking devices, like we do animals, we needed to get data that provided us with millions of movements of individuals. What is amazing about these particular scaling laws is the fact that they are determined by two universal parameters only. This result surprised us all.

In Europe, a similar effort has been underway since 2002 with the introduction of the Euro. On New Year's Day, the notes and coins, all of them valid across the entire euro zone, began to spread across national borders. While the euro is worth the same in every country that uses it, each one mints its own euro notes, with a distinctive design on the reverse and each country introduced its own notes exclusively within its borders. Every time a euro note from Finland appears in Greece, for example, it provides a tiny but precise data point about the relationship between the countries. Moreover, the total amount minted was roughly proportional to each country's economy compared with the overall European economy. So Germany, an economic powerhouse, minted 32.6 percent of all notes, France about 15 percent, and tiny Luxembourg just two-tenths of 1 percent. This Euro diffusion process shows how epidemics spread, to what extent are Europeans integrating and what their travel patterns are.

Dr. Dietrich Stoyan, a statistics professor at the University of Freiberg in Germany in charge of the Euro coin-counting project, comments

I hope that studying this process will help people studying epidemics. What makes this special is that the precise launching date of the coins is known. We know when this `epidemic' broke out.

In contrast to the analysis of the US dollar data, Dr. Stoyan has gone the 'complex' route, by assuming that the relationship between each Euro country depends on a complex set of equations between each country that considers the distance between the countries and the number of commuters, travelers and bank trucks going back and forth etc. His model is composed of 144 interdependent differential equations that take as many of the known variables into account as possible. Another group from the University of Amsterdam, has chosen a high-level model of money flow, based on a branch of probability theory called Markov chains. For example, they assume that a relatively constant percentage of Dutch Euros will leave the Netherlands each month, and that a different, smaller percentage of Dutch Euros that have already left the country will return.

So far, Dr. Stoyan and the Amsterdam group have been surprised to learn that large-denomination coins for one and two euros move much faster than smaller ones. Neither knows exactly why that is. Dr. Stoyan hypothesizes that people tend to dump their small coins out of their pockets at the end of the day and are therefore less likely to take them traveling. Mr. Nuyens believes that the coins are used much more often.

When will the Euro coins reach statistical equilibrium? The models have yielded different results. The Dutch group believes that half of all coins in Holland will be of foreign origin a year from now and that statistical equilibrium across Europe will be reached in five to seven years (from the 2002 launch date).

Questions

• Coin collectors take rare coins out of circulation disproportionately more often e.g. Luxembourg Euro coins are rare relative to German Euro coins. Is this likely to affect the overall conclusions of the Euro 'experiment'? Is there an analagous problem for US dollars?
• Is the self-selected group of participants who log their US dollar bills serial numbers each month a statistically representative sample?
• Science and math teachers have latched onto these projects as a way to illustrate their subjects. They either ask all of the students to examine the change in their pockets or buy rolls of coins at the bank and count them in class. How are the results of these class experiments likely to differ from the web-based experiments?
• Are people more likely to travel further in Summer, when on holidays? If so, might this cause a seasonal effect to emerge in the data and how might an adjustment be made to cater for this?

Further reading

• The scaling laws of human travel, D Brockmann, L Hufnagel and T Geisel, Nature, 26 January 2006 (subscription required).
  

To take part in one of the data gathering exercises:

• See www.eurobilltracker.com/ to take part in the Euro experiment.
• www.wheresgeorge.com/ for the US dollar game.
• And a corresponding Canadian site, called www.WheresWilly.com, after Wilfred Laurier, the man on the Canadian five dollar bill.

Submitted by John Gavin.

Hormone Replacement Therapy (HRT)

Keeping up with turnarounds in the medical field is a full-time occupation. The latest about-face can be found in the Wall Street Journal of January 24, 2006. "In a sharp reversal in thinking about the risks and benefits of menopause hormones, new research shows that early use of the drugs may actually lower a woman's risk for heart disease." An article in the Journal of Women's Health contradicts warnings issued nearly four years ago that hormone use increases risk for heart attack and stroke." The design of that earlier study "didn't necessarily apply to the typical hormone user--women who turn to the drugs to treat menopause symptoms. Most of the participants in the WHI [Women's Health Initiative] were older women who started hormones 10 or more years past menopause." The main points of the latest study:

"Women who began hormone therapy near menopause had about a 30% lower risk for heart disease than women who didn't use hormones."

"Older women who started taking hormones at least 10 years after menopause didn't have any heart benefit."

"Whether the women took just estrogen or a combination of estrogen and progestin didn't make a marked difference in risk."

As the WSJ puts it, "The newest study bolsters the theory that with hormones, timing is everything." Unfortunately, this latest study did not examine other risks--blood clots, breast cancer--of menopause hormones. It also is "not a randomized, controlled trial, proving cause and effect." However, there is current recruitment for a randomized trial known as KEEPS [Kronos Early Estrogen Prevention Study] which "hopes to better answer the question of whether timing of hormone use makes a difference in terms of risks and benefits, particularly to the heart."

Completely absent from the discussion are conjectures concerning strategies for the many women who were on menopause hormones and then stopped when the bad results were publicized. As far as can be ascertained, no studies are underway to shed light on what this considerable number of women should do.

Submited by Paul Alper.

How to count meaningfully

The Undercover Economist, Tim Hartford, FT magazine, 28 Jan 2006
Unblocking business, The Economist, 15-Sep-2005.

Some economists and statisticians spend their careers producing national and international statistics by perfecting theoretical tools and software to gather and analyse noisy data in a messy world. Such datasets are often compiled by counting large samples of things such as goods sold, prices of goods and people out of work. This article higlights how carefully choosing what to count may offer a more simple and transparent alternative quantification of people’s qualitative judgments.

It begins with Chicago-based economist David Galenson investigating who was the greatest artist of the 20th century? He also showed when artists reach their peak by a surprisingly simple approach. He did this by counting something meaningful, which is how frequently experts felt inclined to exhibit Picasso’s art in canonical textbooks. He worked through every major art history textbook of the past 15 years and counted whose art is reproduced most frequently. Picasso, with 395 illustrations in 33 textbooks, scores nearly as many as his three closest rivals (Matisse, Duchamp and Mondrian) put together. From his sample of books, Galenson also observed artists who made conceptual leaps (Johns, Picasso, Duchamp, Warhol) peaked far earlier than so-called “experimental” artists for whom practice made perfect (Kandinsky, Rothko, Mondrian). Johns was 25 when he produced his most-reproduced pieces; Mondrian was 71.

The FT article then switches to a different question but answered with the same methodology.

The World Bank have been counting how many official signatures a farmer in the Central African Republic needs to obtain before he’s able to get his bananas on a ship bound for America or Europe - 38. And how many official procedures must a businessman in Lagos go through in order to legally buy a warehouse - 21.

This kind of counting - done with the help of several thousand local lawyers and public officials - shares common ground with Galenson’s work. It transforms a qualitative impression (”Nigerian bureaucracy is painful”) into a quantitative fact; it does so through the intermediation of experts, and uses a perfectly transparent process. It’s also rather more useful, by showing Central African Republic officials which of the 38 signatures are surplus to requirements in Germany.

The article claims that the resulting publication, Doing Business in 2006 is, by far, the World Bank publication that is most mentioned in press reports. So much so that the Bank's indicators are helping to solve the problems they measure. In the latest report Serbia slashed more red tape last year than any of the 154 other countries in the study and, as a result, start-ups have boomed with the number of registered firms leaping by 42% in 2004. Whereas, in the past, the Bank has been coy about publishing its full league table, for fear of offending the governments at the bottom.

Simeon Djankov, an author of the report, reckons that 21 different reforms over the past two years were inspired by the “Doing Business” audit, which costs just $2m to carry out and disseminate. He says

It is like sports. If you keep score, no one wants to lose.

Questions

• You can see the statistics of your favourite economy for topics like 'Doing Business', 'Hiring and Firing' and 'Paying Taxes'. Are these business-related statistics a good proxy for the state of a whole economy?
• If you were deciding to move to another country what other comparative statistics would influence your decision?

Further reading

Doing Business, The World Bank - "This database provides objective measures of business regulations and their enforcement. The Doing Business indicators are comparable across 155 economies. They indicate the regulatory costs of business and can be used to analyze specific regulations that enhance or constrain investment, productivity and growth."

Submitted by John Gavin.

Beauty is in the eye of the beholder

Mathematician defines beauty in new exhibition, James Randerson, science correspondent, The Guardian, February 1, 2006.
Mathematical photography - beyond beauty, Justin Mullins, New Scientist, 28 January 2006 (Magazine issue 2536).
Photos of mathematical equations -- London show opens Feb 1, Cory Doctorow, BoingBoing blog, 29 Jan 05.

Justin Mullins is a photographer/artist who produces framed equations, with textual material explaining their meaning to everyday life. He is having his first exhibition in London from the 1st to the 12th February at Lauderdale House.

The artist comments

If mathematicians are explorers, then my role is that of a photographer who retraces their steps. During my journey, I photograph what I find. By that I mean frame it, record it and later present it. ... In the same way that an ordinary photograph is a snapshot of an area of outstanding natural beauty, a mathematical photograph is a snapshot of mathematical beauty.

There is an on-line gallery that shows examples (by theme) from the fields of astronomy (starbirth - 'birth', The Three Body Problem - 'family'), physics (entanglement - 'romance', Maxwell's equations - 'let there be light') and, not surprisingly, mathematics (Godel's Incompleteness Theorem - 'mystery', Four Colour Theorem - 'ugly', and Mullins' favourite, Euler's relation - 'beauty'). But, alas, statistical equations did not make it into his selection (t-distribution, anyone?).

The artist invites feedback:

I would be happy to hear how visitors to this site feel about my work. If you have something to add, my contact details are http://www.justinmullins.com/contact.htm.

Questions

• If you were to nominate an equation from the realm of statistics, what would you choose? Most of the equations shown in the on-line gallery also include a description and further links. What text or links would you add to justify your choice?
  • The winner, so far, (based on a very small sample) is the Central Limit Theorem.
• If we consider statical models rather than pure equations, then surely statisticians' most common choice would be a simple regression equation Y = a + bX + e, X and Y are the independent and dependent variables, respectively, a and b are constants and e is a random variable for white noise with a constant variance. A quick check via google returned about 269,000 hits for this statistical model. Can you do better and why?
• The photographs are for sale but at different prices, presumably linked to the size of the frame. What other criteria might be used to evaluate the merits of an equation? For example, if your choice is based purely on statistical grounds, what artistic or historical reasons for your choice might you add?
• Perhaps a better gallery for statistics would be based on graphs rather than equations. What graph would you nominate for your gallery and why? See the R Graph Gallery for some ideas.
• While searching (and failing) to find a snappy title for this Chance entry, I stumbled across two quotes that I can not resist including (can you suggest a catchy title for this item?)

Mathematics, rightly viewed, possesses not only truth, but supreme beauty - a beauty cold and austere, like that of sculpture. Bertrand Russell (1872 - 1970)
Order is the shape upon which beauty depends. Pearl Buck (1892 - 1973)

Further information

• This exhibition runs at Laurderdale House, an arts and education centre based in Waterlow Park in London, from 31st Jan to 12th Feb.

Submitted by John Gavin.

Making a Mint

Run, don't walk to get "Fortune's Formula" by William Poundstone [Hill and Wang, NYC, 2005]. It has everything: Claude Shannon, Gorgeous Gussie Moran, Ivan Boesky, Paul Samuelson, Rudy Giuliani, Michael Milken, Frank Costello and other gangsters. It also has John Kelly's famous formula, based on the even more famous Shannon's channel capacity theorem, for betting on the horses and on the stock market. I also suggest you acquire two reviews of the book: Business Week and [Elwyn Berlekamp's in the American Scientist. See also Chance News 7.

Most statisticians would be at least vaguely aware of Shannon and his channel capacity theorem. According to Berlekamp, " Shannon had shown that noise on a communication channel need not impose any bound on the reliability with which information can be communicated across it, because the probability of transmitting a very long file inaccurately can be made arbitrarily small by using sufficiently sophisticated coding techniques, subject to a constraint that the ratio of the length of the source file to the length of the encoded file must be less than a number called the channel capacity." The notion of a formula for fortune comes about because, " Kelly showed that the asymptotically optimum asset allocation could be determined by solving a system of equations that maximized the log of one's capital. In his horse-track jargon, Kelly also showed that the resulting optimal compound growth rate could be viewed as the capacity of a hypothetical noisy channel over which the bettor was getting the information that distinguished his odds from those of the track. Kelly's betting system, expressed mathematically, is known as the Kelly criterion."

The Business Week review puts Kelly this way: " It's a system that tells you how much of your money to bet to maximize the growth rate of your wealth -- while controlling risk. Technically speaking, it says the fraction of your payroll that you bet should equal your edge divided by the odds. Although Poundstone is a good explainer, the details can be heavy going for casual readers. Here's an example from the book that, for the purpose of simplicity, omits the track's take: The tote-board odds for Secretariat are 5 to 1, so a winning $100 bet pays $600. But you have a tip that Secretariat actually has a 1 in 3 chance of winning, so you have a one-third chance (instead of one-fifth) of winning $600. Dividing $600 by 3 gives you a weighted average payout of $200. Subtracting your bet leaves you with a profit of $100. The edge is defined as the profit divided by the wager, namely $100 divided by $100, or 1. The odds are defined as 5 to 1, or simply 5. So edge/odds equals 1/5, meaning you should bet one-fifth of your bankroll on Secretariat. That's it."

Famous though that Kelly's formula is to the cognicenti, the economists I have spoken to have never heard of Kelly's formula but apparently it was in vogue several decades ago and severely criticized down through the years by Paul Samuelson and other economists who claim it is nonsense; in the eyes of economists, the stock market is "efficient" and is in effect, a (geometric) random walk (prices do not go below zero and thus, the term geometric appended to random walk).

For economists, successful stock pickers are lucky, not smart. On the other hand, according to Berlekamp who was Kelly's research assistant in the early 1960s, hedge fund managers such as Jim Simons, former head of the Computer Science Department at Stony Brook, and Ed Thorp, author of Beat the Dealer and Beat the Market, and others who started out as math professors, are obscenely rich and swear by Kelly. As Berlekamp contemptuously phrases it, "No one who has made a legitimate fortune in the markets believes the efficient-market hypothesis. And conversely, no one who believes the efficient-market hypothesis has ever made a large fortune investing in the financial markets, unless she began with a moderately large fortune." We do not hear about the successful applications of Kelly "presumably because financial wizards as successful as these have always been unwilling to discuss their formulas in public."

Poundstone makes the case that "half Kelly"--or, set aside money from gambling -- is perhaps better because it has "much less volatility" than the straight-forward application of Kelly. Half Kelly or full Kelly, big-time hedge fund managers are hiring mathematicians who can model the system in order to beat the system. The connection to Georgeous Gussie, Milken, Giuliani, etc. makes for lively reading even though Berlekamp decries the "sensationalism." Even if you do not get rich because of the book, the pages will turn faster than you could imagine.

Submitted by Paul Alper.

Dissecting the Line

Dissecting the line.
New York Times, Play Feb. 2006
Stephen Dubner and Steven Levitt

This article by Freakonomics authors Dubner and Levitt is in Play a new sports magazine being published by The Times Magazine and being distributed in the Sunday New York Times. More information about origin of this article can be found at the freakonomics website.

This inaugural issue of "Play" appeared on Freb. 5, the day of the Super Bowl game between the Denver Broncos. In their contribution to this new magazine Dubner and Levitt, inspired by the Super Bowl game, explain how the bookmaker determines the point spread for those who want to bet on the outcome of the game. They describe the point spread as fallows:

The house's bookmaker sets a price in the form of a point spread - the Denver Broncos to win by at least 7 points, for instance. A bettor is then free to take either side. He can bet the Broncos minus 7 points (he'll win this bet if the Broncos win the game by more than 7 points), or he can bet the Broncos' opponent plus 7 points (he'll win if the Broncos lose outright or win by fewer than 7 points; if the Broncos win by 7 exactly, it's a "push," or tie).

Note that this was written before the authors knew who would be playing in the Super Bowl.

The point spread changes as the bets are made and by game time the Steelers were generally considered a 4.5-point favorite so if you bet on their winning you would have won since the Steelers won and the score was 21 to 10.

The authors then explain how the bookies make their money.

The most certain way for a bookmaker to turn a profit is to balance his book - that is, to set a point spread that produces an equal number of dollars wagered on both sides of the line. Since only losers pay the house a 10 percent fee (known as the vigorish, or vig) on top of wagers, a balanced book guarantees the house a 5 percent gain. The conventional wisdom holds that bookmakers set point spreads to achieve this balance.

The authors interviewed Chuck Esosito who runs the race and sports book at Caesars Palace in Las Vegas. Esposito says that is a myth that they can control the spread so that an equal amount is bet on each team.

Dubner and Levitt attribute failure to the fact that the betting public has biases. For example they say, "For every bettor smart enough to stick to home underdogs, there are 5 or 10 bettors who systematically prefer favorites or who underestimate the impact of home-field advantage". The go on to explain how the bookie can and they believe does take advantage of the public's biases.

How does this work? Let's say that a bookmaker is handicapping a game between the Broncos and the Pittsburgh Steelers. He first studies every conceivable element of the game: strengths and weaknesses, momentum, injuries, tendencies, weather forecast, etc. He then decides that the true line - that is, a line that he figures will give each team a 50 percent chance of winning the bet - happens to be Denver minus 7 points. But because of bettor bias, perhaps as much as 80 percent of the money will inevitably flow to the favorite.

So what if the bookie sets the line a little higher, at 9 points? Denver is still likely to draw the majority of the wagering, but its chances of winning the bet are now slightly less than 50 percent. The bookie has thus managed to tempt the majority of the wagering toward an outcome that is unlikely, even if only slightly, to happen. Over time, this pattern will yield the bookie a gross profit margin 20 to 30 percent higher than if he had simply balanced the wagering. In other words, why should a bookie play for the safe 10 percent vig when he can play it only slightly less safe and make much more money?

These claims are supported by a study that Levitt made in his aarticle How do markets function?A empirical analysis of gambling on the national football league. Obviously, it is not easy to get data for such a study but Levitt did get a large dataset that he feels reflects typical football betting.

DISCUSSION

(1) How does a bookies job compare with that of a stock broker?

(2) Which do you think is more likely to make you rich: betting on sporting events or on the stock market?

Submitted by Laurie Snell.