For example, flip a fair coin a million times (a relatively large number of trials), and the probability of heads and tails will be pretty close to 50% for each.

Conversely, the Law of Small Numbers says in a relatively small sample, anything can happen.

That is, flip a fair coin ten times (a relatively small number of trials), and the probability of heads and tails will likely not be 50% for each.

The Law of Small Numbers explains why many people often commit intuitive errors when searching for patterns and relationships in data.

In his book Thinking, Fast and Slow, Daniel Kahneman, psychologist and the 2002 Nobel laureate in economics, offers this interesting example about the incidence of kidney cancer in the U.S.:

CONSIDER THIS: A study of the incidence of kidney cancer in the 3,141 counties of the United States reveals a remarkable pattern. The counties in which the incidence of kidney cancer is lowest are mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West. Now, what do you make of this information?

Your mind has been very active in the last few seconds, and it was mainly operating in what I call System 2 — the “slow” mode of thinking involved in effortful activities such as doing taxes, comparing two washing machines for best value, or driving in traffic. You deliberately searched memory and formulated hypotheses. Some effort was involved; your pupils dilated, and your heart rate increased measurably. But System 1 — the “fast” mode, which includes reacting to loud sounds, understanding simple sentences, and driving on empty roads — was not idle: You probably rejected the idea that Republican politics provide protection against kidney cancer. Very likely, you ended up focusing on the fact that the counties with low incidence of cancer are mostly rural. The statisticians Howard Wainer and Harris Zwerling, from whom I learned this example, commented, “It is both easy and tempting to infer that their low cancer rates are directly due to the clean living of the rural lifestyle — no air pollution, no water pollution, access to fresh food without additives.” This makes perfect sense.

Now consider the counties in which the incidence of kidney cancer is highest. These ailing counties tend to be mostly rural, sparsely populated, and located in traditionally Republican states in the Midwest, the South, and the West. Wainer and Zwerling comment, “It is easy to infer that their high cancer rates might be directly due to the poverty of the rural lifestyle — no access to good medical care, a high-fat diet, and too much alcohol, too much tobacco.” Something is wrong, of course. The rural lifestyle cannot explain both a very high and a very low incidence of kidney cancer.

The key factor is not that the counties were rural or predominantly Republican. It is that rural counties have small populations. And the main lesson to be learned is not about epidemiology, it is about the difficult relationship between our mind and statistics. System 1 is highly adept in one form of thinking — it automatically and effortlessly identifies causal connections between events, sometimes even when the connection is spurious. When told about the high-incidence counties, you immediately assumed that these counties are different from other counties for a reason, that there must be a cause that explains this difference. As we shall see, however, System 1 is inept when faced with “merely statistical” facts, which change the probability of outcomes but do not cause them to happen.

He continues to discuss how humans have a predilection to try to find  predictable patterns in otherwise random events:

A RANDOM EVENT, by definition, does not lend itself to explanation, but collections of random events do behave in a highly regular fashion. Imagine a large urn filled with marbles. Half the marbles are red, half are white. Next, imagine a very patient person (or a robot) who blindly draws four marbles from the urn, records the number of red balls in the sample, throws the balls back into the urn, and then does it all again, many times. If you summarize the results, you will find that the outcome “two red, two white” occurs (almost exactly) six times as often as the outcome “four red” or “four white.” This relationship is a mathematical fact.

A related statistical fact is relevant to the cancer example. From the same urn, two very patient marble counters take turns. Jack draws four marbles on each trial, Jill draws seven. They both record each time they observe a homogeneous sample — all white or all red. If they go on long enough, Jack will observe such extreme outcomes more often than Jill — by a factor of eight (the expected percentages are 12.5 percent and 1.56 percent). No causation, but a mathematical fact: Samples of four marbles yield extreme results more often than samples of seven marbles do.

Now imagine the population of the United States as marbles in a giant urn. Some marbles are marked KC, for kidney cancer. You draw samples of marbles and populate each county in turn. Rural samples are smaller than other samples. Just as in the game of Jack and Jill, extreme outcomes (very high and/or very low cancer rates) are most likely to be found in sparsely populated counties. This is all there is to the story.

Our predilection for causal thinking exposes us to serious mistakes in evaluating the randomness of truly random events. For an example, take the sex of six babies born in sequence at a hospital. The sequence of boys and girls is obviously random; the events are independent of each other, and the number of boys and girls who were born in the hospital in the last few hours has no effect whatsoever on the sex of the next baby. Now consider three possible sequences:

BBBGGG

GGGGGG

BGBBGB

Are the sequences equally likely? The intuitive answer — “of course not!” — is false. Because the events are independent and because the outcomes B and G are (approximately) equally likely, then any possible sequence of six births is as likely as any other. Even now that you know this conclusion is true, it remains counterintuitive, because only the third sequence appears random. As expected, BGBBGB is judged much more likely than the other two sequences. We are pattern seekers, believers in a coherent world, in which regularities (such as a sequence of six girls) appear not by accident but as a result of mechanical causality or of someone’s intention. We do not expect to see regularity produced by a random process, and when we detect what appears to be a rule, we quickly reject the idea that the process is truly random. Random processes produce many sequences that convince people that the process is not random after all. You can see why assuming causality could have had evolutionary advantages. It is part of the general vigilance that we have inherited from ancestors. We are automatically on the lookout for the possibility that the environment has changed. Lions may appear on the plain at random times, but it would be safer to notice and respond to an apparent increase in the rate of appearance of prides of lions, even if it is actually due to the fluctuations of a random process.

Continuing, he exposes the myth of the hot hand in basketball:

AMOS TVERSKY AND his students Tom Gilovich and Robert Vallone once caused a stir with their study of misperceptions of randomness in basketball. The “fact” that players occasionally acquire a hot hand is generally accepted by players, coaches, and fans. The inference is irresistible: A player sinks three or four baskets in a row and you cannot help forming the causal judgment that this player is now hot, with a temporarily increased propensity to score. Players on both teams adapt to this judgment — teammates are more likely to pass to the hot scorer and the defense is more likely to double-team. Analysis of thousands of sequences of shots led to a disappointing conclusion: There is no such thing as a hot hand in professional basketball, either in shooting from the field or scoring from the foul line. Of course, some players are more accurate than others, but the sequence of successes and missed shots satisfies all tests of randomness. The hot hand is entirely in the eye of the beholders, who are consistently too quick to perceive order and causality in randomness. The hot hand is a massive and widespread cognitive illusion.

The public reaction to this research is part of the story. The finding was picked up by the press because of its surprising conclusion, and the general response was disbelief. When the celebrated coach of the Boston Celtics, Red Auerbach, heard of Gilovich and his study, he responded, “Who is this guy? So he makes a study. I couldn’t care less.” The tendency to see patterns in randomness is overwhelming — certainly more impressive than a guy making a study.

He also reveals the real reason behind why some schools appear more successful than others:

I BEGAN WITH the example of cancer incidence across the United States. The example appears in a book intended for statistics teachers, but I learned about it from an amusing article by the two statisticians I quoted earlier, Wainer and Zwerling. Their essay focused on a large investment, some $1.7 billion, which the Gates Foundation made to follow up intriguing findings on the characteristics of the most successful schools. Many researchers have sought the secret of successful education by identifying the most successful schools in the hope of discovering what distinguishes them from others. One of the conclusions of this research is that the most successful schools, on average, are small. In a survey of 1,662 schools in Pennsylvania, for instance, six of the top 50 were small, which is an overrepresentation by a factor of four. These data encouraged the Gates Foundation to make its investment in the creation of small schools, sometimes by splitting large schools into smaller units.

This probably makes intuitive sense to you. It is easy to construct a causal story that explains how small schools are able to provide superior education and thus produce high-achieving scholars by giving them more personal attention and encouragement than they could get in larger schools.

Unfortunately, the causal analysis is pointless because the facts are wrong. If the statisticians who reported to the Gates Foundation had asked about the characteristics of the worst schools, they would have found that bad schools also tend to be smaller than average. The truth is that small schools are not better on average; they are simply more variable. If anything, say Wainer and Zwerling, large schools tend to produce better results, especially in higher grades, where a variety of curricular options is valuable.

The law of small numbers is part of a larger story about the workings of the mind: Statistics produce many observations that appear to beg for causal explanations but do not lend themselves to such explanations. Many facts of the world are due to chance, including accidents of sampling. And causal explanations of chance events are inevitably wrong.

Above quotes excerpted from Thinking, Fast and Slow, ©2011 by Daniel Kahneman. Reprinted in the article The Dangers of Quick Thinking in The Week by permission of Farrar, Straus & Giroux.

The implications of the Law of Small Numbers to evaluating gaming and trading systems are quite clear.  Unfortunately, too many believe testing over hundreds, thousands, and even tens of thousands of trials is sufficient.  Anything can happen in such small sample sizes, and acting on the results of such limited testing is foolish and risky.

Worse are those who don’t even understand the notion of “long term” to appreciate the value in any kind of testing at all.  They seem to regard the tunnel vision required by playing “the shoe at hand” or trading “in the moment” as the heroic way to succeed.  Hopefully they are as oblivious to the pain of loss as they are to the real reason why they sometimes win.

Objective evaluation of a system’s claims should always require testing over sufficiently large data sets which reach into the domain of the Law of Large Numbers, the larger, the better.  When presented with a claim that a system wins 600, 1000, or tens of thousands of shoes or trades, you should not hesitate to ask whether it can win another, and another.

If someone dismisses your request for more statistics, you can be sure one or more of the following are true:  1) he does not understand its importance, 2) he does not know how to go about doing it, 3) he does not want to know, or 4) he does not want you to know.

When money is on the line, none of these reasons are very good.

The hot hand, whether on the basketball court or at the craps table, is "a massive and widespread cognitive illusion" that can be blamed on people being "too quick to perceive order and causality in randomness," says Daniel Kahneman. Photo: Zero Creatives/cultura/Corbis

I don’t know what it is.  Does anyone have an idea what it is?

Please leave a comment, if you do.

The 13-second video I took of the object:

In her study with co-author Dan Ariely, “The Dark Side of Creativity: Original Thinkers Can Be More Dishonest,” to be published in 2012 in the Journal of Personality and Social Psychology, study participants were given aptitude tests, which measured such things as intelligence and creativity.  In other tests, participants had the chance to earn money.  For example, in one test, participants were instructed to transfer their answers onto another test form, but this form had the right answers already indicated, supposedly on accident.  Since their test scores determined how much money they would receive, the situation provided the opportunity to cheat.

Gino found that those who scored high on the tests measuring creativity were also more likely to cheat.  On the other hand, those who scored high only in intelligence and not creativity were not more likely to cheat.

Gino explains in The Boston Globe, “When you’re a creative person, you can use that creativity to come up with reasons for why unethical behaviors may be okay.”

An example of such “self-serving rationalizations,” can be that “other people would cheat under the same circumstances or that a little cheating will not hurt anyone.”

In a Voice of America interview, she summarizes her findings and their implications:

“What we find is that that creativity leads people to be more morally flexible, so they are much more able to come up with justification for the behavior that they’re about to engage in and as a result, they are more likely to cheat.

“We think that creativity really helps people resolve this conflict between something that is more longer term –which is the idea of being good and moral — and then something that is more short term, and is the idea of advancing your own self-interest. And that does not necessarily mean getting money out of cheating, but it could also be getting other types of pleasures or utilities.”

Naturally, creativity and being able to “think outside of the box” are highly valued by our society.  In every aspect of business, management, and government, the top positions which command the most authority and highest salaries are reserved for those who demonstrate the most cleverness and ability to get ahead.

Yet the news is regularly filled with stories of successful and respected bankers, funds managers, and businessmen conning investors out of vast amounts of money, or worse, bringing the entire economy to the edge of collapse.  And, as a group, politicians are not known for being shining examples of ethical behavior in either their private or public lives.  Even scientists have been caught  falsifying their results or plagiarizing others’ work.

Wherever there is money, someone is trying to gain more by cheating, and Gino’s findings suggests he tends to be more creative than his peers.  No level of society is immune from con artists.  Indeed, the petty cons who fill the lower strata are harmless and mere nuisances compared to the real pros at the top who can irreparably damage a civilization.

Gino tries to focus on the positive lesson from her study:  ”So it’s not that we are trying to say that people shouldn’t be creative;  we are trying to say that they should be creative but they should be thinking about the fact that their creativity can be used for the wrong reasons.”

Max has spent years amassing quite an impressive systems library, and he will be posting results of his own testing.

I appreciate the kind words he wrote about me, and he shares the same dedication to objective testing and analysis.

She was literally a “bag lady,” who carried her earthly possessions in a bunch of thin, plastic grocery bags.  Within the bags were basically trash: cans, styrofoam pieces, and other items which appeared to be stuff she had collected, perhaps hoping to trade at a recycling center for a few pennies.  She wore a relatively thin coat with an angular hood, which gave her the vague appearance of a fictional witch.  Her body was slouched over without much support, and her head bent down, as if she were in a state of permanent shame.  From her visage, she appeared to be a white Caucasian in her 50′s or 60′s.

In the priority seating row directly opposite to her sat a heavy set, Hispanic  man.  He wore baggy jeans, several layers of thin coats, a faded orange cap, and a blue band around his right wrist.  He also had a lot of bags, though his bags were of larger capacity and more sturdy.  At first, I did not think he was a homeless man, but he was.  I later learned his name is Steve, and what he did for the lady during the ride was an example of pure selflessness.

At first, Steve simply reached into one of his many coat pockets and pulled out a few extra grocery bags, which he graciously offered to her.  She gratefully accepted.  Then Steve started pulling out all sorts of things one by one  from his pockets to offer her.  After giving her one item, Steve would sit back momentarily, apparently think of another item, reach into his infinitely deep pockets, pull it out, and offer it to her, mostly in silence, but sometimes commenting in perfect English, “This is good,” or asking, “Can you use this?”

Steve gave her bags, a package of sandwiches, a cup of potato salad, a roll of candy, a bag of M&M’s, a bottle of aspirin, a bottle of water, a pair of sneakers, some items of clothing, and even cash (it appeared to be a $5 or $10 bill, but I was too far away to see for sure).

She always accepted his offerings in complete silence, simply reaching out and taking them from his hands, though obviously grateful.  At one point, she seemed to be overwhelmed by his generosity and doubled over with emotion.  She was clearly not accustomed to being showered with such attention and gifts.  She hungrily devoured half of one sandwich, scooped the potato salad with the container’s lid, and munched on the candies for desert.

After he had apparently given her his entire inventory, he sat back, but then roused by another thought, he started taking off his many layers of coats.  ”Here, this will keep you warmer,” he said as he gave her a couple of his coats.  To help her carry all of her new possessions, he gave her one of his larger bags and helped put everything into it.

The oddest item he gave her was an old picture ID card of himself.  ”That’s me, Steve,” he introduced himself.  She studied the card for a few moments, and not quite sure whether to keep it, tried to hand it back to him.  ”Keep it,” he insisted, “that’s for you, too.”  Then after a moment, he laughed, “To help you remember me, your angel.”

Steve then started sharing with her helpful tips about which soup kitchens and churches were offering services and food.  She appeared to listen attentively, and occasionally uttered a few feeble words of acknowledgement, though the loud rumble of the train drowned them out.

At one stop, she looked up while Steve was chatting, grew agitated, quickly gathered up all of her belongings, old and new, got up and hurriedly shuffled to the exit doors.  The doors had already closed, but the train had not yet started rolling forward.  She yanked down hard on the emergency release cable, which automatically opened the doors, allowing her to squeeze in between them to exit the train.

Steve mumbled, “Oh shit, might as well get off here, too,” quickly gathered up his bags, and followed her out.  As he caught up with her on the station platform, he waved to the conductor, “Okay, Driver, you can leave now!”

I was quite moved by the entire experience.  Here was someone who had next to nothing, but selflessly gave what little he had to someone else worse off, who clearly could not possibly offer back anything in return.  It was an act of pure charity.  Steve gave from his relative lack far more than I have ever given from my relative abundance, and I felt greatly humbled by his example.

It is difficult to fully appreciate the circumstances which would reduce someone to such a destitute state of having nothing and homeless on the streets, especially in our modern, supposedly first-world societies.  But while the poor may lack materially, they can be rich in ways which make us more than just human.  The converse is equally true.

To help me remember Steve’s parable of kindness which I was privileged to witness that night, I took a 21 second video of him helping her.  It was the final 21 seconds remaining in the memory card of my camera after the trip, and the footage is one of the most meaningful. I only wish I were able to have captured the entire interaction on video; it would have made a touching video story.

YouTube Video: Steve the Street Man Helping a Fellow Street Woman

Data Sets: Virtuoid 100,000 shoes (ref. My Baccarat Shoe Factory).

Graphs of Net Units Won (after commissions) per Shoe:

High-Hand Card Counting Method:

1) With Money Management::

2) Without Money Management::

Disclaimer: The betting strategies and results presented are for educational and entertainment purposes only. Gambling involves substantial risks, and the odds are not in the player’s favor by design. The author does not state nor imply any system, method, or approach offers users any advantage, and he shall not be held liable under any circumstances for any losses whatsoever.

First, the following table summarizes the previous results of the effects of removing card values on P and B expectancies, excluding ties, giving the numbers of decisions, percentages of total, and Player’s Advantages (P.A., or expectancy) of always betting that decision.  The results suggest that expectancies of decisions in baccarat may significantly depend upon the composition of card values.

Next is a table summarizing card removal effects on whether a decision repeats (R) the last decision, or is the opposite (O) of the last decision, as well as the P.A.s for flat betting repeat and opposite the last decision.  (Shoes are treated as closed universes for the purposes of betting R and O, and thus the first decision of each shoe is not bet upon.)

Note that bias can be effectively measured with repeat and opposite statistics, since decisions in streaky shoes tend to repeat more often than go opposite, and vice versa for choppy shoes.  The above results suggest that even under ideal conditions of complete depletion of certain card values, no significant change in overall bias occurs.  That is, the results indicate that changes in composition of card values do not significantly affect the overall streakiness or choppiness of shoes in baccarat.

To examine the possible effects of card value removal on bias in a different way, B and P event frequencies were comprehensively analyzed for each of the card value removal situations.  The numbers and percentages of events 1 to 26 for B and P, separately and together, are tabulated at the end of this report.  Because of the volumes of data below, a discussion of the results and conclusion are offered first.

As expected, events frequencies are affected by situations favoring a particular decision, for example, generally more frequent longer B events when B is favored.  This naturally makes sense, however, the differences of less than tenths of a percent amount to an event per hundreds to thousands or more events, a measurable but rather useless difference.

Also, for all card removal situations, a slight excess of 1 events exists, for example, the largest difference of 0.553% for the situation in which all odd value cards are removed.  This amounts to an extra 1 event per 180 events.

Besides these extremely small, generally unexploitable effects, depletion of specific card values does not appear to cause significant departures from normally expected event frequencies, and hence, does not significantly affect overall bias.

These results provide objective evidence that overall bias is generally independent of card value composition in baccarat.  It would thus be prudent for those who claim to have viable card counting systems for bias to carefully re-examine the foundations of their approaches for possible systematic errors.

Tables of Percent Differences of Event Frequencies:

To help emphasize the departures from normally exhibited values in a full deck, the following tables provide the differences between the B, P, and total event percentages of each card value removal situation from those of the full deck with all card values present.  Each row is an event, and each column is a card value removal situation.

The original data from which the differences were calculated are given at the end of this report in the section Tables of Comprehensive Events Data.

Percent Differences of B Event Frequencies:

Card Value Removal 0-4 and 2,3,4 and 2,4,6,8 (even):

Card Value Removal 5-9 and 5,6,7 and 1,3,5,7,9 (odd):

Percent Differences of P Event Frequencies:

Card Value Removal 0-4 and 2,3,4 and 2,4,6,8 (even):

Card Value Removal 5-9 and 5,6,7 and 1,3,5,7,9 (odd):

Percent Differences of Total Event Frequencies:

Card Value Removal 0-4 and 2,3,4 and 2,4,6,8 (even):

Card Value Removal 5-9 and 5,6,7 and 1,3,5,7,9 (odd):

Tables of Comprehensive Events Data:

Finally, the effects of card value removal on B and P event frequencies are given.  For each of the card value removal situations, the numbers and percentages of events 1 to 26 for B and P, separately and together, are comprehensively tabulated.

Full decks, no card removal:

Removal of all 0 value cards:

Removal of all 1 value cards:

Removal of all 2 value cards:

Removal of all 3 value cards:

Removal of all 4 value cards:

Removal of all 5 value cards:

Removal of all 6 value cards:

Removal of all 7 value cards:

Removal of all 8 value cards: