CHANCE News 8.05

(28 April 1999 to 7 June 1999)


Prepared by J. Laurie Snell, Bill Peterson and Charles Grinstead, with help from Fuxing Hou, and Joan Snell.

Please send comments and suggestions for articles to

Back issues of Chance News and other materials for teaching a Chance course are available from the Chance web site:

Chance News is distributed under the GNU General Public License (so-called 'copyleft'). See the end of the newsletter for details.

Chance News is best read using Courier 12pt font.


Iacta alea est!

Julius Caesar


Contents of Chance News



Note: The May 1999 issue of Statistical Science will contain a paper: "Solving the Bible Code Puzzle", by Brendan McKay, Dror Bar-Natan, Maya Bar-Hillel, and Gil Kalai. This is a refereed rebuttal to the paper "Equidistant Letter Sequences in the Book of Genesis," by Doron Witztum, Eliyahu Rips, and Yoav Rosenberg (WRR), Statistical Science, Vol. 9 (1994) 429-438.

The paper Solving the Bible Code Puzzle" and additional auxiliary information are available on the web now.

Abe Ross sent this contribution:

The following quote is from the "Focus Section" of the April issue of enRoute, the Air Canada magazine.
An Air Canada flight takes off from YYZ [Toronto] every 4 minutes (on average). An Air Canada flight lands at YYZ [Toronto] every 5 minutes (on average).
Perhaps I am missing something but could this happen unless there were more planes taking off than landing? If so, will not Air Canada eventually run out of planes to take off from Toronto?
We called the enRoute publishers and when said we said were a science writer we did not even have to tell them what our question was. Their mail was heavy. Here is one of the letters they included in the June issue of enRoute along with an explanation.
Destination: Math

In the "Destination Toronto" feature (April), I was fascinated to learn that "on average, an Air Canada flight takes off every 4 minutes and on average one lands every 5 minutes". This means that each hour 15 flights leave, but only 12 arrive and that every day there are 72 more departing Air Canada flights than arriving flights. If there is an average of 100 passengers per flight, you have 2,628,000 more outbound than inbound passengers per year. It is amazing that there is anyone left in Toronto at all.

Greg F. Gulyas, Toronto

EnRoute editor's note:

There is no mass air exodus out of Toronto. The hourly window to land at Pearson International Airport is from 0600 to 0100 hours the next day and the window for takeoffs is from 0630 to 2350 hours. The extra 90 minutes spread over the same number of departures makes it appear that there are more takeoffs than landings.

We used the Air-Canada on-line schedule to try to find a flight outside these intervals but could not. So their time intervals seem correct.


(1) Is it possible for the four numbers we have been given, namely the length of the window for takeoff (17 hours and 20 minutes), the length of window for landing (19 hours), the average spacing between takeoff (4 minutes) and the average spacing between landings (5 minutes) to result in the same number of Air Canada planes landing and taking off in a day?

(2) Why do you think the Air Canada airline has a longer arrival window than departure window?

(3) See if you can determine if your favorite large airport also has different arrival and departure windows.


Election had too many polls and not enough context
The Ottawa Citizen, 5 June, 1999, B7
Harvey Schachter

We do not often see a newspaper article criticizing the way it reports the news but this is such an article. Schachter writes about the way newspapers confuse the public with their tracking of the polls. He starts by commenting that the polls are "crude instruments which are only modestly accurate". The truth is in the margin of error, which is "ritualistically repeated in the boilerplate paragraph that newspapers plunk about midway through poll stories (and the electronic media often ignore)".

He remarks that when the weather forecaster reports a 60% chance of rain tomorrow, few people believe the probability of rain is exactly 60%. But when a pollster says that 45% of the voters will vote for Joe Smith, people believe this and feel that the poll failed if Joe got only 42%. They also feel that something is wrong when the polls do not agree.

Schacter reviews how the polls did in the recent Ottawa election and finds that they did quite a good job taking the margin of error into account--"much better than the people reporting, actually".

In a more detailed analysis of the polls in this election, Schachter gives examples to show that, when newspapers try to explain each chance fluctuation in the polls, they often miss the real reason voters change their minds.

Schachter concludes by saying:

It's amusing to consider what might happen if during an election one media outlet reported all the poll results as a range. Instead of showing the Progressive Conservatives at 46 per cent, for example, the result would be shown as 44-50 per cent.

That imprecision would silence many of the pollsters who like to pretend they understand public opinion down to a decimal point. And after the initial confusion, it might help the public to see polls for what they are: useful, but crude, bits of information...


(1) What do you think about the idea of giving polls as intervals rather than as specific percentages? Would this help also in weather predictions?

(2) Do you agree that weather predictions of the temperature are understood better than poll estimates? For example, what confidence interval would you put on a weather predictor's sixty percent chance for rain?


The numbers game
New York Times, 30 May, 1999
Letter to the editor: Alan M. Dershowitz

Here is the letter:

In his review of John Allen Paulos's "Once Upon a Number" (April 25), James Alexander says that Paulos accuses me of "statisticide" for arguing in the O. J. Simpson case that fewer than 1 in 1,000 women who are abused by their mates go on to be killed by them. He correctly observes that Nicole Brown Simpson was in fact killed, and the issue was whether her alleged batterer was the killer. He then claims, using Bayes's theorem, that "if a man abuses his wife or girlfriend and she is later murdered, the man is the murderer more than 80 percent of the time".

But he fails to note that whenever a woman is murdered, it is highly likely that her husband or her boyfriend is the murderer without regard to whether battery preceded the murder. The key question is how salient a characteristic is the battery as compared with the relationship itself. Without that information, the 80 percent figure is meaningless. I would expect that a couple of statisticians would have spotted this fallacy.

Alan M. Dershowitz
Cambridge, Mass.

The argument used by Dershowitz relating to the Simpson case has been discussed by John Paulos in an op-ed article in the Philadelphia Inquirer (15 Oct. 1995, C7) and his book "Once Upon a Number", by I.J. Good in an article in Nature (June 15,1995, p 541) and by Jon Merz and Jonathan Caulkins in an article in Chance Magazine, (Spring 1995, p 14). See Chance News 4.09, 4.10, and 4.14.

Consider a couple in a relationship. Let M be the event that the women is murdered, A that the man abuses the women, and G that the man is guilty of the murder. Dershowitz states that P(M|A) is very small, say 1/1000. The other authors say this is not the relevant probability because it does not use the fact that you know the women has been murdered. They say the jury would like to know P(G|M and A). The authors of the Chance Magazine article, Merz and Caulkins, estimate this probability to be .8. Dershowitz now argues that the abuse has little effect because just knowing that the women is murdered makes it very likely that it was the husband who did it. Is he correct? Merz and Caulkins write: of the 4936 women who were murdered in 1992, about 1430 were killed by their current or former husband or boyfriend. Thus their estimate for P(G|M) is .29. So they estimate P(G|M and A) = .8 and P(G|M) = .29. Based on this, Dershowitz is wrong. The knowledge of abuse has a significant effect on the probability of guilt.


(1) What do you think Dershowitz was trying to show by his letter?

(2) When discussing this issue in his book ("Reasonable doubts: The O.J. Simpson casse and the criminal justice system", Simon and Shuster 1996) Dershowitz writes:

Prior relationship--with or without batter--is a fairly good after-the-fact indicator of who killed any murdered women. But of course, no jury would ever be allowed to infer that a man murdered his ex-wife just because he had been involved with her.

What do you think about this?


To tell the truth, it's awfully hard to spot a liar
The New York Times, May 11, 1999, F1
Erica Goode

How easy is it to tell when someone is lying? What kind of a test would you design to test the hypothesis that someone is good at spotting liars? This article reports on research performed by Dr. Paul Ekman and his colleagues at the School of Medicine at the University of California in San Francisco. Dr. Ekman, who is a professor of psychology, made videotapes of 10 men, each of whom appears for two minutes and states his opinions on various topics. Some of the men are telling the truth, and some of them are lying. These videotapes are then shown to subjects who are supposed to determine which of the men are telling the truth. The subjects in this experiment included people such as trial court judges, F.B.I. and C.I.A. agents, and trial lawyers. People in these professions might be expected to possess better than average abilities to detect liars.

The results showed that most subjects, including those mentioned above, performed at chance levels or slightly higher. These results have been mirrored by numerous other studies. In a meta- study of such experiments, Dr.Bella DePaulo at the University of Virginia found that in 120 such experiments, only 2 reported subjects' accuracy to be greater than 70 percent.

It is interesting that Dr.Ekman has also found people who regularly score close to 100 percent on such experiments. He has also found that certain occupational groups do better than chance; one such group is the United States Secret Service.

Dr. Ekman points out that in certain situations, one can perform fairly accurately using chance. He gives the example of law enforcement, in which he says that "if you guessed that everybody was lying, you'd be right 80 percent of the time". (Presumably the people to whom he is referring are the suspects, not the police). Although this sounds like a high percentage, he points out that the critical cases are the innocent people who are not lying.


Suppose you were to design an experiment similar to the one carried out by Dr. Ekman. How would you decide if someone "did better than chance"? If you had 10 videotapes, would you tell the subjects in advance how many were lying? In your experiment, does it matter whether your 10 videotapes contain an equal number of truth-tellers and liars?


Ask Marilyn
Parade Magazine, 30 May 1999, p17
Marilyn vos Savant

Are some people just luckier than others? A reader asks:

I've felt for many years that I am a very lucky person. It's not only that I have a good job, good friends and family, but that I am lucky in general. For example, the rain usually waits while I hurry to finish mowing the last piece of grass, even though thunder has been threatening for 10 minutes. Do you believe some people are just luckier than other people? My mom says she is just as lucky. Is luck inherited?

Marilyn replies that she does not believe "that some people are luckier than others, although it may seem that way because even events that are distributed randomly are seldom distributed uniformly. So plenty of people get more or less of this or that". She goes on to attribute the reader's experience to a positive attitude towards life, which she adds may be "inherited" in families.


(1) What does Marilyn mean by saying that randomly distributed events are seldom distributed uniformly?

(2) The apparent tendency of random events to cluster is well- known in probability theory (the famous birthday problem is a classic example). Do you think "luck" is a reasonable way to describe a cluster of happy events? Is Marilyn arguing against this?

(3) Do you think there is a "law of large numbers" for luck?


Eating anger: Blowing your top isn't the same as blowing off steam.
The Boston Globe, 3 May 1999, C1
Judy Forman

Catharsis, Aggression, and persuasive influence: self-fulfilling or self-defeating prophesies?
Journal of Personality and Social Psychology, March 1999
Bushman, Stack, and Baumeister.

The Globe article reports that research reported in the Journal of Personality and Social Psychology contradicts the common wisdom that venting anger is better for you than holding it inside. Brad Bushman of Iowa State University, the lead author on the study, constructed a series of experiments to investigate this. He concludes that venting is "worse than useless...it increases aggression".

Bushman's first experiment involved 360 male and female college students. Half were randomly selected to read a fake news article explaining the beneficial effects of catharsis (i.e., when you are angry, venting your anger in a harmless way will make you less inclined to excessive aggression later). The others read an article explaining that catharsis doesn't work. All then were asked to write an essay expressing their position on abortion. Students were then randomly assigned to receive either positive or negative comments on their essays, with the negative comments designed to provoke anger. In the final step of the experiment, the students were asked to rank ten activities--including hitting a punching bag--they now felt like doing.

Bushman found that those who were not provoked did not want to hit the punching bag, regardless of which article they had read about catharsis. However, of those who were provoked, those who had read pro-catharsis article were twice as likely to want to hit the bag as those who had read the anti-catharsis article.

In a second experiment, 600 male and female students were again randomly assigned to read articles pro and con. This time, all of them were given negative comments on their essays, and they were then given the chance to actually hit a punching bag. Seven women declined, but all the others hit the bag! Each subject was then paired with an "opponent" (one of the investigators) for a competitive task. The subject had the opportunity to behave aggressively by blasting the opponent with a loud noise. The students who read the pro-catharsis article were reportedly "twice as aggressive" as the others. The more they had liked hitting the bag, the more aggressively they now behaved. Thus the previous venting had apparently not defused their anger.

In a final experiment, 100 students read pro-catharsis messages and were then provoked. This time, instead of hitting the bag, they were told to sit quietly for two minutes prior to the competitive task. Their subsequent behavior was much less aggressive than that of the students in the previous experiment. Bushman concludes that there is something beneficial in the old advice of counting to ten before expressing anger.

Other studies have shown that strong emotions like anger can trigger heart attacks. Dr. Murray Mittleman, of the Institute for the Prevention of Cardiovascular Disease at Boston's Beth Israel Deaconess Medical Center, directed a 1995 study of more than 1600 heart attack survivors. He found that the risk of heart attack more than doubles in the two-hour period after an experience of intense anger. The article adds that the heart attack effect described by Mittleman actually represents a rather small risk. A healthy 50-year-old man has a baseline risk for heart attack of one-in-a-million in any hour, so anger raises this to only two-in- a-million.

This article is a good example of the need to look at the original article and to compare what is reported against what actually is claimed in the article.

The article suggests that it is catharsis itself that is shown to not be effective. In the original article the authors state that studies have already shown this and, indeed, some studies have shown that the act of venting your anger in a harmless way can lead to more aggression rather than less.

What the authors of the study are interested in is why this catharsis theory is still so generally believed. They ask if the reason could be because catharsis is so often claimed to be effective in the media. This could lead to a kind of placebo effect, or a self-fulfilling effect, which makes it work. The aim of their research is to see if this is true. They find that it is not.

The Globe article makes no mention of the effect of the media in the discussion of the study. For example,

In this article, as in the other news articles about this study, we read:

The research, published recently in the Journal of Personality and Social Psychology, shows that catharsis - verbal or physical venting - is "worse than useless," says the lead author, Iowa State University psychologist Brad J. Bushman.

In the original article we read in their conclusion:

Our findings suggest that media messages advocating catharsis may be worse than useless.

You can find a review of the studies on catharsis in Gee,, R.G., & Quanty, M.B. (1977), "The Catharsis of aggression: An evaluation of a hypothesis. In L. Berkowitz (Ed.), Advances in experimental social psychology (Vol. 10, pp. 1-37) Academic Press NY.

As the Globe article reports, there have been studies showing that anger can be bad for your health. You can find a summary of this research in the book "Human Aggression" by Russell Geen, Brooks/Cole, 1990.


(1) In the description of the findings from the second experiment, what does it mean to say that the readers of the pro- catharsis article were "twice as aggressive"? How do you think the investigators measured the relationship between degree of aggression in competition and the degree to which subjects liked punching the bag?

(2) Regarding Mittleman's study, what about those victims who don't survive their heart attacks?

(3) Does it seem important to you to discuss baseline risks? How often to you see them reported in medical news stories?

(4) Why do you think that the press did not mention the true objective of this study.


The Spring issue of Chance Magazine arrived with lots of interesting articles.

Chance magazine has revived its "Chance Musings", which are reviews of articles from the press. The selection in this issue are particularly interesting and, surprisingly, there is very little overlap with our own choices--which suggests that there are many more interesting articles out there that neither of us catch. We mention three articles that we particularly enjoyed but all the articles are excellent.

Searching for the "Real" Davy Crockett, by David Salsburg and Dena Salsburg. p. 29

This article gives an account of the Salsburg's attempt to determine the authorship of books and writings attributed to Davy Crockett. Davy Crockett, besides being a T.V. star in the 50's, was a real person and a member of the U.S. Congress during the 1830's. Crockett is reported to have written three books: A Narrative of the Life of David Crockett. Written by Himself, An account of Col. Crockett's Tour to the North and Down East, and Col. Crockett's Exploits and Adventures in Texas. In addition some of his speeches in Congress have been preserved in Gales' and Seaton's Register of Debates in Congress. Evidently, historians have questioned whether the book by Texas was written by the same person that wrote the other two books.

The authors use the method developed by Mosteller and Wallace and used to settle disputes about the authorship of the Federalist papers. This method involves looking at the distribution of a set of "contentless" words. The Salsburg's conclude that the writer of the Texas book was indeed not the same person who wrote the Narrative and the Tour books. They also conclude that the same person who wrote the Narrative wrote the congressional speeches and this was the real David Crockett. They looked at two likely candidates for the Texas story--James Fenimore Cooper and Nathanial Hawthorne--and were able to reject the hypothesis that either of these authors wrote the Texas book, leaving the author of this book a mystery.


The Mosteller-Wallace method concentrates on looking at the use of contentless words such as: upon, also, an, by, although, enough, while, etc. In determining that Joe Klein was the author of Primary Colors, Donald Foster asked: what other author who wrote about Clinton regularly used uncommon words such as hawkish, puckish, lugubrious, etc., that were used by the writer of Primary Collers. What are the advantages and disadvantages of these two different methods?

Then Penalty-Kick in Soccer: Does it Make Sense to Shoot at the Keeper?. by Jack Brimberg and Bill Hurley. p. 35

An article by Jack Brimburg and Bill Hurley uses game theory to determine if it ever makes sense for the penalty kicker in soccer to shoot at the keeper. That the kicker should do this was suggested by their observation that the keeper often tries to anticipate where the kicker will hit the ball and is on the move when the ball is kicked. The authors model this situation as a two person game between the kicker and keeper and find the optimal strategy for this game. This optimal strategy suggests that the kicker should aim at the keeper about 20 percent of the time. The authors suggest that the embarrassment of aiming at the keeper on an occasion when he does not move might explain why this strategy is not used as often as they believe it should be.

A Statistician Reads the Sports Pages, by Scott M. Berry, Column Editor. How Many Will Big Mac and Sammy Hit in `99? p. 51.

Scott Berry boldly estimates before the season begins who the top 25 home-run hitters will be this year. He does this by estimating a distribution for the number of home runs that each of the leading contenders will achieve. Taking the expected value as an estimate we find that Mark McGwire ranks first with 59, followed by Ken Griffey with 51 and Sammy Sosa with 50.

Scott assumes a binomial model for the number of hits for a given number of at-bats with the probability of getting a hit, theta, itself a random variable with a beta distribution. He estimated the parameters for the beta distribution using the players' performances in previous years. He assumes that the number of at-bats is a random variable with a normal distribution.

We decided to see how his predictions are doing as of June 9th. Here is what we found:

Rank Player HR At-bat Rate HR At-bat Rate Rank
Pred Pred Pred 6/9 6/9 6/9 6/9
1 Mark McGwire 59 515 8.7 18 182 10.1 5
2 Ken Griffey 51 602 11.8 22 215 9.8 1
3 Sammy Sosa 50 618 12.4 20 220 11.0 3
4 Juan Gonzalez 44 603 13.7 16 197 12.3 9
5 Vinny Castilla 39 604 15.5 12 211 17.6 31
6 Albert Belle 39 592 15.2 10 208 20.8
7 Jose Canseco 39 566 14.5 22 211 9.6 1
8 Andres Galarraga 39 550 14.1 out
9 Rafael Palmeiro 37 595 16.1 17 198 11.6 6
10 Greg Vaughn 37 515 13.9 13 177 13.6 19
11 Mo Vaughn 37 585 15.8 13 164 12.6 19
12 Manny Ramirez 35 568 16.2 14 218 15.6 13
13 Barry Bonds 35 528 15.1 4 41 10.3
14 Jeff Bagwell 34 552 16.2 17 202 11.9 6
15 Jim Thome 35 518 14.8 8 163 20.4
16 Alex Rodriguez 34 650 19.1 10 95 9.5
17 Vladimir Guerrero 34 602 17.7 13 210 16.2 19
18 Jay Buhner 34 524 15.4 6 82 13.7
19 Mike Piazza 34 558 16.4 11 171 15.5
20 Nomar Garciaparra 33 626 19 10 199 19.9
21 Larry Walker 33 548 16.6 13 170 13.1 19
22 Carlos Delgado 33 239 7.2 14 226 16.1 13
23 Tino Martinez 32 575 18 9 208 23.1
24 Henry Rodriguez 32 506 15.8 11 153 13.9
25 Frank Thomas 31 561 18.1 5 201 40.2

The first column is the expected value of Scott's distributions for the number of home runs. Scott also gives the standard deviations for these distributions, and these range from 7.3 for Thomas to 10.2 for McGuire. With the exception of Vinny Castilla and Albert Belle, the top players from Scott's list are doing very well as predicted. We have added the home run rate prediction which is the number of at-bats per home run--small is good.

Comparing the predicted rates and the rates so far this season, we note that the majority of the players are doing better than predicted by Scott. The New York Times ("More runs and high scores", 23 May 1999 by Murray Chass) discussed the fact that home hitting was up 15% from last year and people are already trying to figure out why. The usual explanations of "expansion" and "the ball is livelier and smaller" are brought out. However, a new and more interesting explanation is that the strike zone was altered this season by the commissioner's office. (See the discussion question)


According to the New York Times article:

Umpires were instructed before the season to stop calling strikes on pitches off the corners of the plate. In recent years, the strike zone had spread horizontally. At the same time, umpires were told to call strikes on pitches two inches above the belt. For many years, pitches above the waist had been called balls.

What effect do you think this change would have on batting averages, home runs hit, and runs batted in?


A number of news articles covered the World Series of Poker. We selected four articles that discuss this event from different points of view and made this our feature topic for this Chance News.

$1M Poker King Noel trumps the world to land a fortune
The Mirror 15 May 1999
Neil Leslie

A straight face
Newsweek, 17 May 1999
Seth Stevenson

Poker's house gets fuller; Good ol' boy game drawing new faces
Boston Globe, 15 May, 1998
Mark Matassa

Playing your cards right; poker comes out of the back room and
into the computer science lab
Science News, 18 July 1998
Ivers Peterson

The annual World Series of Poker (WSOP) was held at Binions Casino in Las Vegas from May 10 to May 13. These four articles discuss this event from different points of view.

The Mirror article describes this year's World Series of Poker. The Globe article describes the demographic changes in the participants in recent years' WSOP. Increasingly, women enter the WSOP. Also recent series have seen more young, relatively inexperienced poker players with a good knowledge of probability and game theory who challenge the seasoned poker players who rely on years of experience and an ability to read the kind of hand their opponents have. The Newsweek article has wonderful interviews with professional poker players explaining how they can get information about the other players' hands by watching for player's involuntary movements, expressions, etc. Finally, the Science News article discusses the prospects for a computer one day winning this famous poker tournament.

One of the players in this World Series was Jeff Norman who provided our Chance Lecture "Hedge Funds and Gambling in Capital Markets". In his earlier days Jeff made his living playing professional poker and he still participates in the World Series. He helped us understand how the game series is played. Jeff made a good showing, ending up 83rd in the field of 393 players.

The game played in the WSOP is called Texas Hold'em. To appreciate the subtleties of this tournament it helps to know how Texas Hold'em is played and how the tournament is run.

The players pay $10,000 for the privilege of playing and for competing for prizes, the largest of which is $1 million. They start with $10,000 worth of chips. This year there were 393 players resulting in about $4 million dollars worth of chips among the players. The players put in an ante at the beginning of each game. This ante and the minimum bet allowed are raised as the game proceeds. There is no limit to the size or the number of bets that can be made. Players play at tables with 9 or fewer players. When a player loses all his chips he or she is out of the tournament. The bets become sufficiently large that this happens at a pretty good clip. Play continues on the second day until there are only 36 players left. Those forced out have enjoyed some exciting poker but leave $10,000 poorer. Play continues on the third day until there are only 6 players left and on the fourth day these six players play until there is only one player left, the champion.

It is important to bear in mind that the approximately $4 million dollars in chips that the players start with remains in the game even as the number of players decreases. Thus, in the final game two players with about $4 million dollars in chips compete for the championship. When the winner is determined, the $4 million dollars (more accurately $3,930,000) that the players paid to play is given as prizes to the 36 players who made it to the 3rd day. The prizes start at $1 million dollars for the champion, then $768,625 for the runner-up and continue to decrease down to $15,000. Thus everyone who makes it to the third day goes home a winner.

Here is how Texas Hold'em is played at the WSOP. It is a high- card-only game. Players end up with seven cards from which they make a five card hand. The best such five-card hand wins the pot. In case of a tie the pot is divided.

Players put into the pot an ante. This ante increases through time ending up at $3,000 for the final hands. One player is designated as the "dealer", indicated by a button at his position. The player to the left of the dealer is called the "first blind" and the player to his left is called the "second blind". The blinds are required to make bets which, together with the ante, provide an incentive for players to stay in the game.

Two cards are dealt face down to each player. The first blind makes a mandatory bet (initially $25) and the second blind makes a mandatory bet twice as large (initially $50). Like the ante, the amount of the blind bets increase as play progresses until, at the end, these bets are $15,000 and $30,000. The second blind bet is always twice the first blind bet. The player to the left of the second blind is first to act (called "under the gun" because it is a disadvantage to act first). He can either fold, call the big blind by putting in the same amount, or raise any amount he likes. Play continues around the table clockwise until all players have either folded or put in the initial bet plus all the raises. When the play returns to the blinds for the first time, they can use their initial bets as part of the money needed to call or raise.

When there are no more raises, three cards are dealt face up. These are called "community cards" and can be used by all of the players to make their final five-card hand. This is followed by another round of betting. When this betting is finished, a fourth card (called the "turn") is dealt face up and another round of betting takes place. Finally, a fifth community card (called the "river") is dealt face up followed by a final round of betting. At the end of this round, players still in the game who want to claim the pot show their cards. The best five-card hand, formed from the players' two cards and the five community cards, wins the pot. The pot is divided if there is a tie.

If a player does not have enough chips to match a bet, he can put in all his chips and competes with the other players for the amount of his chips. For example if someone bets $5000, and you only have $2000 left, you can call for all your chips. If another player calls the $5000, then the dealer will place $2000 x 3 = $6000 in the main pot, which is all you will be eligible for. The remaining chips are put in a side plot and the other players compete for these chips. When one hand is completed the button and the two blinds then move one position to the left and the next hand is played.

To play professional poker a player needs to have a good knowledge of, or intuitive feeling for, the probability that future cards drawn will make a competitive hand. A player also has to have an ability to make an intelligent guess as to the kind of hand his opponent might have and whether he is bluffing or not.

As an example to show the interplay of probability and psychology, consider the final play of the series. The two remaining players were: Noel Furlong, a 61-year-old Irishman who manufactures carpets, races horses and makes large bets on them, and Allan Goehring, a 36-year-old New York corporate bond trader. When his hand was played, Furlong had about $3 million in chips and Goehring a little less than $1 million. The blinds had been increased to $15,000 and $30,000 and the ante to $3,000. When there are only two players, the player with the button, in this case Furlong, is also the first blind. Here is the description of the final hand as given by Leo Muzer, writing for Poker Digest.

Furlong limps in on the button. (We assume this means that he put in $15,000 to call the $30,000 second-blind bet). Goehring gently taps the table. The deal flops QH QC 5S. Both players check quickly. Alan checks when the deuce of spades turns. Noel thinks for five seconds, then bets $150,000. Alan quickly comes over the top with a $300,000 raise. Noel watches Alan's hands as he shoves in the raise. Now Noel looks at the board for 10 seconds. He looks down at his massive amount of chips and announces, "I'm going all in". The crowd rises as Alan counts his chips (he has $600,000 left). He pushes his chips in creating the largest pot in the 30-year history of this great event. Noel carefully places the five of hearts and five of clubs in front of him. The crowd explodes realizing the big Irishman flopped a full house and trapped his opponent. Alan's shoulders sag and he flips black sixes onto the board. He has two chances---a queen or a six. There's no miracle on the river. It's the eight of spades.

As we have remarked, the decisions made by the player make use of both psychology and the players' knowledge of probability theory. In our discussion questions we have asked you to think about how the players in this last hand made the choices they did. Jeff Norman has agreed to critique any analyses that are sent to us. A prize will be given for the best analysis.

The issues involved in bluffing and in trying to read an opponent's hand are discussed in the Newsweek article by Seth Stevenson. Stevenson tours Binions with legendary poker players who explain the signals that weaker players give out and the way that professionals try to control their own involuntary reactions. The technical name for a clue about the other players hand is a "tell". Here is a typical discussion in this article:

At the Bellagio poker room, topflight player Mori Eskandani takes me on a tour spanning all levels of play. In low-stakes games, weak players show classic tells-- when they get good cards they reflexively touch or look at their chips, just raring to go. But up in the high- stakes room, the pros, playing for thousands of dollars a hand, are a blank slate. There are about 400 great professional poker players in the world, says Eskandani, and among them you won't spot 10 tells. Reading them is a much more subtle game.

Of course top players will also fake tells. Andrew Glazer tells of the time he had a "monster" hand and wanted his opponent to bet.

He stared at me, and I wasn't making a move, so finally I gave a tiny gulp, like I'd had to gulp the whole time and just couldn't hold it. He instantly bet, and I won. Of course, a better player might have known I was faking and folded right away.

The Globe article discusses the new wave of young poker players. One such player is Darse Billings, a former computer science graduate student at the University of Alberta in Edmonton who now plays poker for a living. Billings chose the development of a poker-playing computer program as his thesis topic. He found that understanding the theory and mathematics of poker gave him a solid foundation for becoming a serious poker player. He continues to consult with researchers at the University of Alberta who, under the direction of Jonathan Schaeffer, are developing a computer program to play Texas Hold'em.

The Alberta poker program is called Loki--named after the Norse god of mischief and chaos. While still under development, Loki already plays a strong game. It employs a probabilistic strategy for each choice the player has, check, fold, call, or raise. When Loki faces a decision it invokes a simulator to estimate the expected value of each possible decision. The simulator generates an instance of the missing information, subject to any constraints that have been learned, and then simulates the rest of the game to help it make a decision. Of course, it is easy to introduce bluffing and other strategy decisions into its play and Loki will not be making any nervous movements which give away its play.

You can read a detailed account of how Loki works and how it is being tested in a technical report: Using Probabilistic Knowledge and Simulation to Play Poker by Billings, Pena, Schaeffer and Szafron available on the web.

Loki can be played on the web using the Internet Relay Chat (IRC) poker server. It is a consistent winner against this sample of players but no doubt they are a far cry from the players that Loki will face when it enters the WSOP.

To get an idea what it is like to play in the World Series of Poker Jeff recommended one of two books: Big Deal, by Anthony Holden and The Biggest Game in Town, by Anthony Alvarez. These books are out of print but your library might have one of them (ours did). Of the two, Jeff said that the Alvarez book is the more interesting. To learn about poker strategy, he suggests reading the classic book Poker Theory, by David Sklansky. We also found it useful to play a few games by downloading the demo.


(1) Why do you think Furlong checked on the flop (after the first 3 cards were dealt)?

(2) Analyze Goehring's decision to call or fold on the end.

Send us your answers to these two questions and compete for the grand prize as judged by WSOP player Jeff Norman.


Correction: Reader Eric Foster pointed out that Prof. John Lamberth, who did the studies on profiling discussed in Chance News 8.04, is at Temple University rather than Trinity University as we stated.

Chance News
Copyright © 1998 Laurie Snell

This work is freely redistributable under the terms of the GNU General Public License as published by the Free Software Foundation. This work comes with ABSOLUTELY NO WARRANTY.


CHANCE News 8.05

(28 April 1999 to 7 June 1999)