Sunday, 1 April 2001


You may have come here,

Seeking the rules,

In which case you’ve fallen,

For Cassini’s April Fool’s

Thursday, 1 February 2001

Navinder Singh Sarao – The Hound of Hounslow

It took Navinder Singh Sarao a long time to accept that he might have been scammed out of $50 million. Stuck in London’s Wandsworth prison, wracked with anxiety and unable to sleep, the realization dawned on the man dubbed the “Flash Crash Trader” as slowly as spring turned to summer outside the barred window of his jail cell.

The trauma of the past few weeks had been difficult to process. On April 20, 2015, the slight, doe-eyed 36-year-old had dozed off peacefully in the same suburban bedroom he’d slept in since he was a boy. The next day he was arrested and taken to a police station, where he was charged with 22 counts of fraud and market manipulation carrying a maximum sentence of 380 years.

According to the U.S. government, the British day trader had made tens of millions of dollars using an illegal practice called spoofing, including, fatefully, on the morning of May 6, 2010, when the Dow Jones Industrial Average fell almost 1,000 points in minutes before bouncing back. The extent of Sarao’s culpability for the flash crash is fiercely contested, but the incident exposed the shaky foundations on which the hyper-fast, computer-dominated financial markets now rest.

Sarao’s bail was set at 5.05 million pounds ($6.3 million). It was a hefty sum, but according to the accounts of his company, Nav Sarao Futures Limited, he’d earned 30 million pounds in the previous five years. Newspaper reports, in which Sarao was dubbed “The Hound of Hounslow,” speculated that he’d be back with his family in the shabby West London borough by the weekend. 

Instead, the nightmare got worse.

Where’s the money, Nav, his lawyers wanted to know. Sarao couldn’t make bail, they gradually learned, because the bulk of his wealth was tied up in investments and offshore trusts, each more complicated than the last. Days in Wandsworth prison, a Victorian-era fortress where Sarao was housed with sexual predators and violent offenders, turned into weeks.

After four months of dead ends, his legal team struck a deal with the authorities: If the U.S. Justice Department and the Commodity Futures Trading Commission agreed not to oppose a reduction in bail to 50,000 pounds, the firm would act as a bounty hunter, taking on responsibility for tracking down the missing millions on the condition that its fees be paid if it did.

They were going down a rabbit hole. A review of Sarao’s investments from 2005 to the present day, based on dozens of interviews and thousands of pages of documents, reveals another twist in an already remarkable story. 

Navinder Sarao, the trading savant accused of sabotaging the world’s financial markets from his bedroom, may himself have been the na├»ve victim of what his lawyers portray as a series of cons that stripped him of almost every cent he earned.

Sarao declined to comment for this article. His lawyer, Roger Burlingame of Kobre & Kim in London, told a U.S. judge in November that all of the defendant’s assets “have been stolen.” Sarao invested in ventures from which he, the law firm and the CFTC had been unable to recover the funds, Burlingame said. “Basically, he has some extraordinary abilities with respect to pattern recognition and certain sorts of mathematical abilities, but he has some fairly severe social limitations.”

Sarao’s trading career started inauspiciously in 2002 at Futex, a fledgling outfit in an unglamorous office an hour from the City of London that housed wannabe traders in exchange for as much as 50 percent of their profit. In a roomful of recent college graduates and drifters, Sarao stood out from the pack.

“Nav was always going to be the kind of person that would be legendary in some way,” Futex Chairman Paolo Rossi said in an interview with Bloomberg TV after Sarao’s arrest. He had “the potential to be remembered as one of the world’s greatest traders.”

It wasn’t until Sarao left Futex in 2008 and struck out on his own that he started to make serious money. Public filings show his assets popped to 14.9 million pounds from 461,000 pounds in the 12 months ending in June 2009, long before he enlisted a programmer to build a system that authorities say was designed to cheat the market.

Former colleagues talk about Sarao’s frugality—his scruffy clothes, his reluctance to spend money on cars and watches, his abstemious eating habits. He learned early at Futex that withdrawing cash ate into his bankroll and reduced the size of trades he could place.

That near-obsessive drive to hold on to as much of his wealth as possible can also be seen in the way he conducted his business affairs. Looking to minimize his tax bill, he was introduced by his accountant to John Dupont, a director at the London arm of an Isle of Man-based financial advisory firm called Montpelier Tax Consultants.

Dupont, then in his mid-30s, was a high-energy salesman whose accent veered from upper class gent to Guy Ritchie cockney depending on who he was speaking with, a former employee recalls. Operating from an office on Cockspur Street in London’s West End, members of his team cold-called contractors, day traders and bankers and tried to enlist them in a range of plans to minimize their tax bills, documents seen by Bloomberg show.

The aim was to identify loopholes before they were closed. One former Montpelier employee said he coaxed wavering customers to sign up by promising to pay their legal bills in the event of a clampdown by Her Majesty’s Revenue and Customs. Everyone at the firm thought he was Alec Baldwin in “Glengarry Glen Ross,” the person said.

Self-employed traders were particularly good prospects because they were predisposed to high levels of risk. And Sarao, an absent-minded dreamer with an unerring gift for making money who would later be diagnosed with Asperger syndrome, would prove to be the ultimate mark.

In 2009, on the advice of Montpelier, Sarao entered into a complicated dividend-stripping scheme that resulted in a major reduction in his tax bill, according to a close adviser to Sarao who spoke on the condition of anonymity. Happy with the result, Sarao went a step further the following year, the person said.

Among Dupont’s crew was Miles MacKinnon, a polished so-called introducer who had left school for a stint as a rugby player before heading to the City of London. For four months in 2010, MacKinnon became the only other director of Sarao’s firm.

Around the same time, Sarao set up two employee benefit trusts in the Caribbean island of Nevis, according to a document filed in Sarao’s case. He plowed his earnings into those trusts, then gave himself interest-free loans to trade with and live on, the adviser said. The arrangement meant Sarao all but avoided paying corporate taxes. One vehicle was named the “NAV Sarao Milking Markets Fund.”

Dupont and MacKinnon said in an e-mail that they “did not introduce or advise” on the Nevis trusts.

Sarao had an uncanny ability to attract controversial characters. He sought advice from tax specialist Andrew Thornhill, who in 2015 would be charged by the British barristers’ industry group with five counts of professional misconduct. And, as the Wall Street Journal reported, one of Sarao’s trusts was, for a period, affiliated with David Cosgrove, the Irish director of Belvedere Management who has been barred by Mauritius authorities from serving as a company officer because of regulatory violations. Thornhill declined to comment. Cosgrove didn’t respond to e-mails.

In 2011, the British government ended the benefit-trust gravy train. Sarao paid back the loans and restructured his business. Montpelier was investigated and dissolved, and about 3,000 of its customers were ordered by a judge to pay 200 million pounds in back taxes. Fraud charges against two directors were later dropped.

MacKinnon and Dupont—along with a third partner, Ryan Morgan—then founded MacKinnon Dupont Morgan, which was later reborn as MD Capital Partners. The firm describes itself on its website as a boutique private equity firm.

They leased an office in Mayfair, home of hedge funds, Michelin-starred restaurants and private members clubs. MacKinnon joined the Worshipful Company of International Bankers and the executive board of the Special Olympics. He and Dupont set up about a dozen companies between them, focusing on industries such as renewable energy.

Dupont and MacKinnon said in their e-mail to Bloomberg that they “never made, or introduced investments to projects that are purely driven by tax breaks” and that at the time they got involved in renewables there weren’t any tax incentives in place. Morgan, who left the firm, didn’t respond to a request for comment.

By 2011, Sarao had trebled his assets to 42.5 million pounds. He agreed to become an investor in an Isle of Man-based entity called Cranwood Holdings, set up to acquire land in Scotland that would one day house wind farms, according to two advisers to Sarao. Documents on the enterprise filed in the British dependency are light on detail, but the advisers say Sarao put about 12 million pounds in Cranwood—money they say Dupont and MacKinnon could access.

Dupont and MacKinnon said in their e-mail that Sarao conducted “substantial independent due diligence” before investing in Cranwood and that he approved all of its payments. One of their companies, Wind Energy Scotland, is funded by and provides project management services to Cranwood.

The pair also acted as agents for more exotic ventures, such as sending divers to search shipwrecks for sunken treasure. The returns on offer were never less than impressive. Sarao, who told acquaintances he harbored aspirations of becoming a billionaire, invested in several. All were tame compared with what came next.

Sometime in 2012, Sarao was introduced—again through Dupont and MacKinnon—to a squat, intense Mexican named Jesus Alejandro Garcia Alvarez, who was looking for investors for his company IXE Group. Garcia said he was the scion of a family of billionaire landowners and industrial-scale farmers with swaths of land around the world. He had arrived in Zurich from Latin America a few years earlier and had been working hard to build a reputation ever since.

IXE was conceived as a one-stop shop for high-net-worth individuals, offering services ranging from asset management to event planning to advice on private schools. Then, around the time Sarao met Garcia, the company’s website underwent a radical overhaul. Gone were the concierge services. IXE was henceforth a “conglomerate of companies worldwide” involved in “agribusiness, wealth management, commodity trading and venture capital.”

Articles appeared in the Swiss media profiling the mysterious young man making waves among Zurich’s business elite, including pictures of Garcia wearing a poncho over his suit, arm outstretched across Bolivian salt plains he said he owned. One newspaper put him on its annual rich list. Garcia was invited on Bloomberg TV to talk about his family’s quinoa interests, then on CNBC to discuss the “white gold rush” for lithium.

Garcia had all the trappings of a successful entrepreneur: half a dozen sports cars, a small but well-appointed office in the center of Zurich, a glamorous Russian wife. He even joined the Swiss board of the Robert F. Kennedy Center for Justice & Human Rights, an organization whose U.S. directors include Tim Cook and Martin Sheen.

Garcia flew to London and met with Sarao two or three times, according to people with knowledge of the matter. In an interview on IXE’s website, Garcia laid out his pitch to investors: “We are offering alternative investment vehicles that provide constant returns to investors. The investment in real economy makes the advantages obvious—investors are benefiting from constant returns generated from actual transactions with zero speculation and zero volatility.”

Garcia told Sarao he would get an annual 11 percent return, the people said, and assured Sarao that any money he handed over would be used only as collateral, not put at risk. He also introduced Sarao to Swiss banking contacts, they said. The trader was again restructuring his business, this time around an Anguilla-based vehicle called International Guarantee Corporation.

Sarao did some due diligence about IXE, according to one adviser, but he seems to have overlooked a few red flags: The company website is littered with spelling mistakes, and several executives are members of Garcia’s family.

Garcia initially agreed to meet to discuss this story, then opted to respond to questions through a colleague at IXE. The colleague, Dominic Forcucci, wrote in an e-mail that Garcia hadn’t done anything improper and that IXE “properly disclosed the risks of investments” to Sarao. A lawyer representing Garcia, William Wachtel, later said that Garcia described any allegations against him as “baseless and without merit.”

On Aug. 20, 2012, documents show, Sarao agreed to give about $17 million to Garcia and his company—by far his biggest investment and a substantial chunk of his net worth. He later invested an additional $15 million, according to a person with knowledge of the matter. Even though they’d met on only a handful of occasions, he would describe Garcia to associates as a friend.

Sarao may have been particularly trusting, but he wasn’t alone in buying into the IXE miracle. Former employees interviewed by Bloomberg describe Garcia as charming and, on first meeting, impressive. He offered commissions to third-party agents to send prospective investors his way, ensuring a steady stream of business and creating a buzz around the firm.

In 2014, Garcia signed a deal to acquire Banca Arner, a Swiss lender in decline after allegations that it had helped former Italian Prime Minister Silvio Berlusconi hide money. To coincide with the transaction, Arner’s new marketing chief, Garcia’s wife Ekaterina, issued a press release announcing it had appointed a new chairman: Michael Baer, a great grandson of the founder of private bank Julius Baer Group and a respected figure in Swiss banking.

IXE just needed sign-off by Switzerland’s financial regulator, Finma. In order to seal the deal, Finma told Garcia he’d have to come up with 20 million Swiss francs ($18.7 million) in capital and account for where it came from. After heated meetings with the regulator and the owners of Arner, Garcia offered to hand over the money in unmarked gold, according to two people with knowledge of the talks. Without a stamp, the gold was unacceptable to the regulator, and in the end Garcia walked away from the deal, leaving Baer and a raft of other new recruits frustrated and embarrassed, the people said. Baer and a spokesman for Finma declined to comment.

For the time being, though, Sarao had no cause for concern. IXE sent him periodic statements showing the interest accruing in his accounts. As ever, he was happy to let it sit there and grow.

By then, Sarao’s readiness to consider almost any opportunity that offered an attractive rate of return was well-established. After another strong year in 2013, Dupont and MacKinnon introduced him to Damien O’Brien, a physically imposing Irish entrepreneur with aspirations to revolutionize the online-gaming industry.

The unique selling point of O’Brien’s company, Iconic Worldwide Gaming, according to a pitch document seen by Bloomberg, was that it allowed gamblers to bet on movements in currencies and securities using an interface that looked like an online casino, with a roulette wheel and buttons for “higher” and “lower” instead of red and black. The patented software was called MINDGames, short for Market Influenced Number Determination games.

The concept may not have pleased Gamblers Anonymous, but the financial projections were enticing. O’Brien predicted in the pitch document that Iconic would go from a standing start to a cash balance of 110 million pounds by the end of its third year. There were also some reassuring names on the board: Robin Jacob, a U.K. appeals court judge, and David Michels, a former deputy chairman of Marks & Spencer.

In July 2014, documents show, Sarao invested 2.2 million pounds in Iconic. Cranwood Holdings extended loans of an additional 1 million pounds, according to one Sarao adviser. He was, several times over, the largest investor in the company.

Dupont and MacKinnon said in their e-mail that Sarao was an experienced gambler and trader who conducted his own due diligence on the gaming sector before investing. They also said they objected when Sarao told them he planned to lend money to Iconic. O’Brien didn’t respond to requests for comment. Jacob and Michels said they were no longer board members.

In the months following Sarao’s investment, O’Brien went on a campaign to increase Iconic’s profile. The company sponsored World Touring Car Championship driver Rob Huff and filmed a slick advertisement with mixed martial arts superstar Conor McGregor. O’Brien and his employees were photographed ringside or wining and dining clients. In one shot taken in Las Vegas and posted on Twitter, a line of promo girls posed in matching uniforms with Iconic logos emblazoned on their hot pants. In another, O’Brien stood next to a matte-black Rolls-Royce with the license plate DAMI3N.

By the time Sarao was arrested in April 2015, he had about $50 million tied up in investments around the world, according to people with knowledge of the matter who even now aren’t positive it’s all accounted for. It was only as his lawyers tried to recoup the money that he was forced to face up to the possibility that it was gone. Sarao was released that August after his parents put up the family home as collateral against the bail of 50,000 pounds.

In November of last year, following an unsuccessful extradition fight, Sarao flew to Chicago where he pleaded guilty to one count of wire fraud and one of spoofing, which entails placing bids or offers with the intention of canceling them before they’re executed. He was ordered to pay $38.4 million to the CFTC and the Justice Department, which determined that, of the money he made by day trading, only $12.8 million came from cheating the market.

Sarao is scheduled to find out the length of any custodial sentence later this year. In the meantime, he has been allowed to return to Hounslow, where he is banned from trading and, despite pushing 40, placed under the care of his father.

Sarao’s lawyers are no closer to getting their hands on the money beyond about 5 million pounds seized from his trading accounts after his arrest. The CFTC and the Justice Department have joined them in the hunt, according to people close to the situation. The agencies could try to compel banks holding Sarao’s assets to give them up, but that might not be easy because most of the money is outside the U.S. Spokesmen for the CFTC and the Justice Department declined to comment, as did Burlingame, a former Justice Department prosecutor who represents U.K. targets in U.S. investigations.

IXE told Sarao it would return the cash in instalments in 2015 and 2016, according to a person familiar with the matter. The deadlines came and went, but no money has been produced. Garcia is rarely seen driving his sports cars around Zurich anymore, according to former associates. In October, German magazine Brand Eins skewered what it portrayed as his outlandish claims about plots of land in Bolivia and Mexico and linked Garcia to Burton Greenberg, who’s serving eight years in a Florida prison for fraud.

Former IXE employees interviewed by Bloomberg say that Garcia spent whatever he brought in to fund his own lavish lifestyle and that projections he gave in presentations to Sarao, Baer and others were plucked out of thin air. Still, Garcia’s efforts to acquire a bank continue. In August, IXE announced it was buying Private Investment Bank in the Bahamas from Swiss firm Banque Cramer & Cie. The deal is scheduled to be completed this month.

Garcia hasn’t been accused of any wrongdoing. Forcucci, the IXE spokesman, said the company is “working to return the money in a fair and equitable manner to its investors.”

Iconic went into liquidation in January 2016. A company hired to advise it on resale options said O’Brien had underestimated the cost of breaking into the online gaming market by about 10 million pounds.

Sarao’s lawyers have been unable to retrieve his investments in Cranwood despite repeated requests, owing to its convoluted offshore ownership structure, according to a person with knowledge of the situation. Dupont and MacKinnon said in their e-mail that Wind Energy Scotland has been working to get funds to Cranwood.

From their base in Berkeley Square, the pair last year started another company focused on renewable energy, Celtic Asset Management, which offers “access to a substantially higher return profile, with less capital at risk.”

Meanwhile, Sarao is back in his bedroom. The computer that got him into so much trouble is gathering dust in a Washington evidence room. Depending on how much the authorities are able to recoup, he will probably spend the rest of his life paying back the money he owes. If they really want it, they could always lift the trading ban, one associate quips: He’d make it back in no time.

Monday, 1 January 2001

Bundeslayga System

In September 2010, I noticed that the strategy of laying home favourites in the German Bundesliga was consistently profitable, making this an ideal system for mitigating Betfair's Premium Charges.

The rules are simple. Selections are those teams priced by Pinnacle at between 1.3 and 2.00.

In the three full seasons of 2012-15, (prior to this there is no Pinnacle price data) backing the home favourite in the top European Leagues is a profitable strategy with an ROI of 3.93% from 434 bets, but laying home favourites in Germany has been profitable in each of those three seasons.

Because the lay option is only available on the Exchanges, and Pinnacle do not offer a 'lay' price, the formula used in my calculations adjusts the home price depending on the Pinnacle over-round.

For example a Home price of 1.51 in a Pinnacle Sports market of 102% is adjusted to 1.5395.

Here are the results (updated to include 2015-16):

** Note that from 2016-17, the range has been adjusted. Please see this New Bundeslayga post for details.

New Bundeslayga

In September 2010, I noticed that the strategy of laying home favourites in the German Bundesliga was consistently profitable, making this an ideal system for mitigating Betfair's Premium Charges.

The basic Bundeslayga System layed teams priced by Pinnacle at between 1.3 and 2.00.

The results for the four seasons 2012-2016 are posted here, but with the addition of Pinnacle's closing prices to Football Data's spreadsheets, I took a fresh look at the approach and decided to tweak it for a couple of reasons. 

One was that the System was losing some of its potency - markets adapt, and my feeling is that much of the value of laying home odds-on selections was now gone (possibly in part my own fault for publicising this system).

A second reason was to determine whether the 1.3 to 2.0 range was still optimal. Eliminating shorties and odds-against selections made it a simple system, but reviewing the closing price data shows that the implied profitability range of 40% to 75% is more profitable.

The table below shows the results of the Traditional Bundslayga and for the New Bundeslayga, using Pinnacle's Closing prices.
I've left the original returns using the early scrape Pinnacle prices on this post which shows the difference that price movements can have on a system.
Note that because the lay option is only available on the Exchanges, and Pinnacle do not offer a 'lay' price, the formula used in my calculations adjusts the home price depending on the Pinnacle over-round.

For example a Home price of 1.51 in a Pinnacle Sports market of 102% is adjusted to 1.5395. 

1 / [( 1 / 1.02 ) * ( 1 / 1.51 )]  

UMPO System

UMPO is an acronym for Underdogs MLB Play-Off and is a simple strategy which backs odds-against underdogs in the Major League Baseball Play-offs.

Since 2004, (the first year for which I have data) it has been a steady winning strategy, and the following returns from a US site can generally be beaten on the exchanges making this an ideal strategy for mitigating Betfair's Premium Charges.

Updated to end of 2016 season. 

BLUnders System

BLUnders (Big Line Unders) is a neat little system that I came up with in the Summer of 2015 when I was asked my opinion on NBA Totals betting in games where one side was heavily favoured.

My gut response was that Unders was more likely to offer value, my thinking being that a team with a comfortable lead has a tendency to psychologically ease up, rest starters, and in addition the often crucial extra points from the fouling towards the end of a close contest are absent.

I looked at the data for the last twenty seasons to see if this idea had any merit, and found that while the older data from 1995-2004 showed more equity between Overs (474) and Unders (480), the numbers since 2005 showed more of a bias (54.8%) towards Unders (487 - 402) and since 2010, even more of a bias at 56.5% (247 - 190).

The number of qualifiers is a little under 100 per season, a very manageable number for a part-timer, although the 2015-16 season has seen an increase with the dominance of the Golden State Warriors and San Antonio Spurs having an effect.

One additional note here is that the system is even more profitable in the play-offs. In the 20 completed seasons since 1994, the Unders has a 15 - 10 advantage.

** Update after 2015-16 season. 

For the fifth consecutive season, the system returned a profit, albeit lower than the previous four seasons.
Most qualifying matches are, as might be expected, home games, but on the rare occasions (less rare over the past two seasons) when it is the Away team favoured, the results are:
The second system I play alongside the BLUnders mentioned previously meant that the combined totals for the season ended up as below:

Not the sexiest of profits perhaps, but as I have said before, their value lies in the churn volume and the ease with which selections are determined.


While the basic concept is easy to grasp and it sounds very easy to do, in fact trading has developed into quite a complex and tricky art, because people have designed bots and software to take advantage of the most obvious situations where trading should give you a profit. In addition there are people out there who manipulate the markets using “spoof” money, large sums which appear suddenly in an attempt to move the market in a particular direction. 

These can disappear as fast as they appear, but it is often not easy to tell the difference between spoof money and real money until it is too late. When trading you are swimming in shark-infested waters and you need to have your wits about you or you will get eaten alive.

[Courtesy of Juicestorm]


The p-value is the level of marginal significance within a statistical hypothesis test representing the probability of the occurrence of a given event.

The p-value is used as an alternative to rejection points to provide the smallest level of significance at which the null hypothesis would be rejected.

A smaller p-value means that there is stronger evidence in favour of the alternative hypothesis.

Read more: P-Value Definition | Investopedia

Benford's Law

Benford's law, also called the first-digit law, is an observation about the frequency distribution of leading digits in many real-life sets of numerical data. The law states that in many naturally occurring collections of numbers, the leading significant digit is likely to be small. 

For example, in sets which obey the law, the number 1 appears as the most significant digit about 30% of the time, while 9 appears as the most significant digit less than 5% of the time. 

By contrast, if the digits were distributed uniformly, they would each occur about 11.1% of the time. 

Benford's law also makes (different) predictions about the distribution of second digits, third digits, digit combinations, and so on.

It has been shown that this result applies to a wide variety of data sets, including electricity bills, street addresses, stock prices, house prices, population numbers, death rates, lengths of rivers, physical and mathematical constants,and processes described by power laws (which are very common in nature). It tends to be most accurate when values are distributed across multiple orders of magnitude.

It is named after physicist Frank Benford, who stated it in 1938, although it had been previously stated by Simon Newcomb in 1881.

The Gambler's Fallacy And Law Of Large Numbers

The Gambler's Fallacy & law of large numbers by Mirio Mella

The Law of Large Numbers was established in the 17th century by Jacob Bernoulli showing that the larger the sample of an event - like a coin toss - the more likely it is to represent its true probability. Bettors still struggle with this idea 400 years on which is why it has become known as the Gambler’s Fallacy. Find out why this mistake can be so costly.

The Law of Large Numbers

Using a fair coin toss as an example (where the chance of hitting heads and tails has an equal 50% chance), Bernoulli calculated that as the number of coin tosses gets larger, the percentage of heads or tails results gets closer to 50%, while the difference between the actual number of heads or tails thrown also gets larger.

It’s the second part of Bernoulli’s theorem that people have a problem understanding – which has led to it being coined the “Gambler’s Fallacy”. If you tell someone that a coin has been flipped nine times, landing on heads each time, their prediction for the next flip tends to be tails.

This is incorrect, however, as a coin has no memory, so each time it is tossed the probability of heads or tails is the same: 0.5 (a 50% chance).

Bernoulli’s discovery showed that as a sample of fair coin-tosses gets really big – e.g. a million – the distribution of heads or tails would even out to around 50%. Because the sample is so large, however, the expected deviation from an equal 50/50 split can be as large as 500.

This equation for calculating the statistical standard deviation gives us an idea what we should expect:

0.5 × √ (1,000,000) = 500

While the expected deviation is observable for this many tosses, the nine-toss example mentioned earlier isn’t a large enough sample for this to apply.

Therefore the nine tosses are like an extract from the million-toss sequence – the sample is too small to even-out like Bernoulli suggests will happen over a sample of a million tosses, and instead can form a sequence by pure chance.
Applying Distribution in betting

There are some clear applications for expected deviation in relation to betting. The most obvious application is for casino games like Roulette, where a misplaced belief that sequences of red or black or odd or even will even out during a single session of play can leave you out of pocket. That’s why the Gambler’s Fallacy is also known as the Monte Carlo fallacy.

In 1913, a roulette table in a Monte Carlo casino saw black come up 26 times in a row. After the 15th black, bettors were piling onto red, assuming the chances of yet another black number were becoming astronomical, thereby illustrating an irrational belief that one spin somehow influences the next.

Another example could be a slot machine, which is in effect a random number generator with a set RTP (Return to Player). You can often witness players who have pumped considerable sums into a machine without success embargoing other players from their machine, convinced that a big win must logically follow their losing run.

Of course, for this tactic to be viable, the bettor would have to have played an impractically large number of times to reach the RTP.

When he established his law, Jacob Bernouilli asserted that even the stupidest man understands that the larger the sample, the more likely it is to represent the true probability of the observed event. He may have been a little harsh in his assessment by once you have an understanding of the Law of Large Numbers, and the law (or flaw) of averages is consigned to the rubbish bin, you won’t be one of Bernouilli’s ‘stupid men’.

Law Of Small Numbers

The Law Of Small Numbers By Joseph Buchdahl 

The law of small numbers is a cognitive bias where people show a tendency to believe that a relatively small number of observations will closely reflect the general population. Read on to test your logical powers with the hospital quiz and find out how graphs can be misleading and what you can do to avoid losses when using stats to place your bets.

The Hospital Quiz

In 1974 two psychologists, Daniel Kahneman and Amos Tversky, presented their experimental subjects with the following scenario, accompanied by a question. A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day and in the smaller hospital about 15 babies are born each day.

As we know, about 50% of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower. For a period of one year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days?

The larger hospital
The smaller hospital
About the same (within 5% of each other)

According to binomial theory, the number of days where boys born outnumber girls by at least six to four will be nearly three times greater in the smaller hospital compared to the larger one, simply on account of the larger volatility in birth ratios. A larger sample is less likely to stray very far from 50%. Yet only 22% of respondents gave the correct answer.

What are heuristics?

Kahneman and Tversky described this error as a belief in the law of small numbers. More generally, judgements made from small samples are often inappropriately perceived to be representative of the wider population. For example, a small sample, which appears randomly distributed, would reinforce the belief that the wider population from which the sample is selected will also be randomly distributed.

Conversely, a small sample demonstrating an apparently meaningful pattern – such as nine heads from 10 coin tosses – will cause the observer to believe that the population will display the same meaningful pattern. In this case the assumption would be that the coin is biased. The experience of perceiving patterns in random or meaningless data is called apophenia.

A belief in the law of small numbers is part of a wider group of mental short cuts that people take when making judgements under uncertainty. Kahneman and Tversky called these short cuts heuristics. Making generalisations from small samples is an example of a representativeness heuristic, where people assess the probability of a particular event based solely on the generalisation of previous similar events that comes easily to mind.

Another example of the representativeness heuristic is the expression of the gambler’s fallacy. Indeed, such a bias arises out of the belief in the law of small numbers. As Kahneman and Tversky say:

The heart of the gambler's fallacy is a misconception of the fairness of the laws of chance. The gambler feels that the fairness of the coin entitles him to expect that any deviation in one direction will soon be cancelled by a corresponding deviation in the other. Subjects act as if every segment of the random sequence must reflect thetrue proportion; if the sequence has strayed from the population proportion, a corrective bias in the other direction is expected.

Reading graphs of unequal sample sizes

Sports bettors can be particularly prone to faulty pattern recognition through a misplaced belief in the law of small numbers. Misinterpreting profitability from small samples of wagers as representative of a departure from randomness and evidence of predictive skill can have unpleasant financial consequences over the longer term. Consider the hypothetical profitability chart of 100 wagers on NFL point spreads below. Each bet is struck at a price of 1.95. Impressive, isn’t it?

What if I told you this record comes from a well-known US sports handicapper? With a decent growth trend and a yield of 15% you might be forgiven for believing me. Of course, I’m lying. In fact, the next chart of 1,000 wagers reveals the bigger picture.

Really there was no long term profitability to be had at all. The reason: this was merely produced by a random number generator which assumed a 50% chance of an individual win and a profit expectation of -2.5%. The first chart simply represents the initial 100 wagers of the second.

Yet even in the second longer time series a healthy profitability was maintained for several hundred wagers. Furthermore, despite showing an overall loss, the pattern of the time series looks anything but random, with a fairly consistent wave-like pattern to it.

However, as Kahneman and Tversky recognised, we are far more likely to perceive sequences of similar outcomes as being non-random even if there is no underlying mechanism behind them. Of the two binary sequences below, which looks random and which not?

0, 0, 0, 0 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1

0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1

The majority of people would pick the second sequence. In fact, the first was generated randomly in Excel and I made up the second purposely with shorter sequences of 1s and 0s. When asked to create random sequences like this many of us will switch from 1 to 0 or vice versa if we feel that one of them is happening too often.

Now take a look at the following 1,000-wager charts. They were all randomly generated. The large range of possible outcomes should provide you with a flavour of just how easy it is to be fooled by apparently meaningful patterns.

Remember, these are not series of 100 wagers, but 1,000. Take a look at the middle one. It has all the hallmarks of an expert tipster or bettor with a 5% yield and solid profit growth throughout the entire sequence of betting, the sort of performance the best handicappers are capable of long term. And yet it happened just by chance.

Using the binomial distribution we can work out the probability of still being in profit after a period of betting despite having an expectation of -2.5%.

Number of Wagers (odds 1.95, 50% win probability) / Minimum number of wins needed / Probability of being in profit

100 52 38.22%
250 129 32.90%
500 257 28.05%
1000 513 21.46%
2500 1283 9.68%
5000 2565 3.40%
10000 5129 0.51%

After 1,000 wagers we still have over a 1-in-5 chance of being in the black despite our betting being nothing more than random. If we placed one handicap bet on every NFL game played, this would take us nearly four seasons. That’s a long time to believe we have anything other than luck on our side.

How small is small?

The law of small numbers is a cognitive bias where people show a tendency to believe that a relatively small number of observations will closely reflect the general population. Furthermore, as this exercise has shown, small can sometimes be quite large. It exists because people favour certainty over doubt, explanation over ignorance, causation over association, patterns over randomness and skill (particularly self-serving skill) over chance. For sports bettors, failure to truly appreciate its significance can be costly.

MLB: T-Bone System

The T-Bone System (named after the person who came up with the original idea) is a baseball (MLB) system backing road favourites (-140 or hotter) coming off a loss, and introduced in 2016.

Here are the results over the past five seasons, updated to the end of 2016.

2016 was the best recent season overall, with all three bet categories (Money Line, Run Line and Overs) in profit. 

Expected Value (EV)

Expected value (EV)

In probability theory, the expected value (EV) of a random variable is the weighted average of all possible values a random variable can take on.

The expected value may be intuitively understood by the law of large numbers: It can be interpreted as the long-run average of the results of many independent repetitions of an experiment (e.g. a dice roll).

Example: If you roll a dice, the possible outcomes are 1, 2, 3, 4, 5 or 6 – all with equal probability of 1/6. The expected value of a dice roll is 3.5.

This example shows that the "expected value" is not a result that may be "expected" in the ordinary sense. Rolling a 3.5 with a dice (or having 2.5 children) is impossible.

In trading, we can speak of EV as the estimated value of an investment with unknown return.

A simple example on EV in trading

We can buy an apple for $10. We expect the following (based on analysis or on our experience):
A likelihood of 50% that we can sell the apple for $16.
A likelihood of 25% that we can sell the apple for $12.
A likelihood of 25% that we will not be able to sell the apple and it goes bad.

The expected revenue of the "apple deal" would be:

(50% x $16) + (25% x $12) + (25% x $0) = $8 + $3 + $0 = $11.

Thus, the expected value of the transaction would be $11 minus the $10 that we spend for the apple. The resulting EV of $1 is positive, indicating that doing the deal would be profitable, or "+EV".

The return on investment in this scenario would be +10%.

In many situations, the expected value of an investment can be estimated with thorough analysis.

Positive result versus positive expected value

Differentiating between a positive result and a positive expected result is a key skill needed for every trader.

A trade that made you profit might in fact have had a negative expected value. And a trade that made you a loss might in fact have had a positive expected value.

Coming back to the dice example: you bet $100 on the result of a single dice roll being larger than 2. Obviously, this is a good bet. Still, the result might be a 1 – and you lose the bet.

EV and money management

It also is obvious that you should not bet your life on the dice roll, even if your bet has a positive expected value. The potential loss simply is far too big. This is a good example that shows that you should not take any trade or bet, even if you have a positive expected value.

This is one of the fundamental ideas behind money management. Do trades with a positive expected value, but do not invest more than 1-2% into a single trade. Otherwise, the risk of going broke despite making the right decisions simply is too large.

Lay Last Loser

The Lay Last Loser System is used in the top five English Football Leagues, and lays the last team to lose in each division until they win again. The idea was found on a post on the Betfair Forum and suggested excluding the Premier League and laying teams in their next two games. I have modified this to include the English Premier League and to keep laying teams until they win again.

The reasoning appears to be that once a team loses, the market rates them higher than the team might rate themselves! Confidence is powerful, and a loss can quickly make a serious dent in it.

The system is complete for the 2015-16 season, with the final summary as below:

The 2016-17 season saw this system take a small loss:

Lay System Running Totals 2009-10

League : Bets - Wins ROI%

Bundeliga: 10 - 9 +43.46

.Ligue 1 : 9 - 10 -1.8

.....EPL : 16 - 12 +53.8

Last Updated: 18.Sep.2010

The Smart Money: How the World's Best Sports Bettors Beat the Bookies Out of Millions

A Book of Gambling Secrets, but Some Are Easily Spilled

Published: November 11, 2004

In the world of gambling, secrets do not last long, and it is not always easy to tell when someone is bluffing.

Those two maxims could weigh on the minds of editors at Simon & Schuster over the next year as it prepares to publish "The Smart Money: How the World's Best Sports Bettors Beat the Bookies Out of Millions."

The book is said to be the true story of the Brain Trust, a group of professional gamblers that legally wagers hundreds of thousands of dollars in Las Vegas each week on college and professional football games, making it one of the most influential forces in the gambling world.

The book's author was a participant in the group for seven years, first as a courier betting the syndicate's money, then as the operator of his own, smaller gambling ring affiliated with the Brain Trust. The proposal identifies him only as 44, an identity adopted, the author said, to protect the privacy of some of the participants in the gambling ring.

Books about gambling have been hugely popular in recent years, and the proposal for "The Smart Money" reads like an edge-of-the-seat thriller, one made all the more intriguing by the likelihood that it is true. David Rosenthal, the publisher of Simon & Schuster's flagship imprint, and Marysue Rucci, a senior editor there, agreed two weeks ago to buy the book for an advance of roughly $500,000. Publication is scheduled for October.

Simon & Schuster hopes to add an extra element of suspense by having the author wager some of his advance on the number of copies of the book sold in the first year: exceeding a certain number wins 44 a bonus, while falling short of the mark means the author has to return some of his advance.

But both the publisher and the author face several potential pitfalls. While all gamblers lie sometimes, changing the identifying details in a purportedly nonfiction book leaves readers in the uncomfortable position of not knowing which parts are really true. And it turns out that the supposedly anonymous players in this gambling drama are, in fact, very well known. A few hours of Googling and database searching by a reporter led to the real identities of 44, the Brain Trust and its principal character. The author is Michael Konik, a freelance writer who worked for several years for Bill Walters, a Las Vegas developer who is one of the country's biggest sports bettors.

The book proposal identifies the leader of the gambling ring as Rick Matthews, describing him as "the kingpin of American sports betting," a philanthropist, restaurateur and Southerner who, by dint of being "one of the greatest golf hustlers of all time," is "a millionaire several times over."

Those descriptions make it relatively easy to determine that Rick Matthews is actually Mr. Walters, a Kentucky native who now spends much of his time developing golf courses in Las Vegas. In a telephone interview, Mr. Walters, whose gambling history has been explored many times in magazine and newspaper profiles, confirmed that he is Rick Matthews.

For years Mr. Walters has run the Computer Group, a betting syndicate that uses computers to analyze reams of information on sports teams and players, placing enormous bets when it determines the point spread in Las Vegas books is out of line with its own calculation. Mr. Walters has been indicted at least three times in federal or Nevada state courts on charges of illegal gambling, but the indictments have been dismissed each time; he has never been convicted of any gambling charge. In 1999 The Las Vegas Review-Journal named him as one of the 10 most influential nonparticipant figures in sports.

Mr. Walters and others also said 44 was Mr. Konik, a television commentator on poker matches for Fox Sports who has written extensively about gambling and golf. Among his several books is "The Man With the $100,000 Breasts and Other Gambling Stories" (Huntington Press, 1999), which includes a profile of a high-stakes golf hustler who is a thinly disguised Mr. Walters.

Mr. Konik, reached at his home in Los Angeles, declined to discuss whether he was 44. Mr. Rosenthal, the publisher, also said he had no comment about Mr. Konik's relationship to 44.

In an earlier telephone interview, arranged by his literary agent, Jennifer Joel of International Creative Management, the anonymous 44 sounded much like Mr. Konik. In the interview, the author said all the events to be described in the book were true and were based on an extensive journal kept during his tenure at the gambling group, from 1996 to 2003. According to the proposal, the author recorded in the journal his gambling wins and losses, brushes with the law and "the stunning sensation of actually holding $1 million worth of cash in my hands."

But the author said he believed that he needed to change identifying details of people in the book because many of the participants in the gambling ring, as well as the peripheral characters in the casinos and elsewhere, are not public figures. As such, their abilities to sue successfully for libel or defamation are greater than they would be if they intentionally sought publicity.

The author also said that while all of his gambling activities were legal, not everyone who knows him is aware of his gambling venture, and he would rather keep it that way. The details revealed in the book proposal, however, seem unlikely to allow that to happen. The proposal details the author's start-up of his own gambling circle, which he calls "the Hollywood boys" and which includes some well-known but as-yet-unidentified celebrities.

Mr. Rosenthal, the publisher, said he did not think that the book needed an anonymous author to succeed. But so far, at least, he said he was willing to honor the author's desire to remain unknown. "He's not a C.I.A. station chief, but he has some specific reasons for wanting to keep his identity under wraps," Mr. Rosenthal said. "It's important to him, which I respect. But I think I could do it either way."

Michael Konik went along for a gut-crunching ride as Big Daddy beat the bookies

Reviewed by Jeff Ostrowski
December 31, 2006

Anybody who has played the office football pool knows how easy it is to fancy yourself an expert handicapper, and just how hard it is to outsmart the Las Vegas line. The matchups sit there like ripe fruit to be plucked. How can the unbeaten Bears be mere 10-point favorites against the lowly Dolphins? Then Rex Grossman throws three interceptions, the Bears lose three fumbles, and the Dolphins win easily.

Add in the bookies' 10 percent commission – which means you have to be right more than half the time just to break even – and you get an idea why casinos are so profitable.

Michael Konik knows these gut-wrenching ups and downs all too well. The magazine writer spent four football seasons as part of a gambling syndicate, an experience that taught him two lessons: Yes, you can outsmart the bookies. And no, it's not worth it.

Not unless you possess ironclad nerves and a bottomless well of patience. You had also better know a programmer who can whip up an algorithm to pick the winners, because opinions and hunches make for sucker bets.

Konik's introduction to this clandestine world came in 1997, when he met Rick “Big Daddy” Matthews. The legendary sports gambler recruited the scribe to work as one of the dupes who placed bets on behalf of Matthews' Brain Trust.

Matthews needed the help because every bookie in Las Vegas had refused his business. So Konik showed up at Caesar's Palace and concocted a story painting him as a high roller who'd cashed a big advance for a screenplay. In fact, he was simply moving Matthews' money.

But there was no such person as Rick “Big Daddy” Matthews, no such syndicate as the Brain Trust; both are names Konik made up. The New York Times and Las Vegas Review-Journal have reported that Konik's boss actually was Las Vegas developer and gambler Billy Walters, and the syndicate was known as the Computer Group.

Konik floats through the first-season honeymoon in which he is entrusted with briefcases full of cash, gets a front-row seat on the action and enjoys the high-roller treatment at Vegas' poshest casinos. The money isn't bad, either. As one of Big Daddy's operatives, Konik keeps 10 percent of the winnings but risks none of the losses.

But he operates under a constant cloud of suspicion: The bookies suspect that he works for Matthews, and any time he launches a winning streak, they limit his bets or close his account. Konik's girlfriend, tired of his gambling obsession, moves out.

By 1999, with offshore casinos booking millions through the Internet, Konik moves his money online, but he finds that the bookies in Costa Rica and Antigua are even slipperier than his pals in Vegas.

Konik deftly builds suspense throughout his fast-paced account, and it's not giving away too much to say that he finally decides that he's just not cut out to be another Big Daddy.

“All the time and psychic energy I devote to wagering on sports surely could be better spent on something else,” he writes in an uncharacteristic bout of introspection. “Anything else.”

© Cox News Service

Joseph Jagger - The Man Who Broke The Bank At Monte Carlo

Joseph Hobson Jagger (1830 – 1892) was a British engineer, widely known as The Man Who Broke the Bank at Monte Carlo, though he is not the only person to have done so. His name is sometimes reported as Jaggers, but the International Genealogical Index indicates that Jagger is more likely.

Jagger was born in September 1830 in the village of Shelf near Halifax, Yorkshire. Jagger gained his practical experience of mechanics working in Yorkshire's cotton manufacturing industry. He extended his experience to the behaviour of a roulette wheel, speculating that its outcomes were not purely random sequences but that mechanical imbalances might result in biases toward particular outcomes.

In 1873, Jagger hired six clerks to clandestinely record the outcomes of the six roulette wheels at the Beaux-Arts Casino at Monte Carlo, Monaco. 

He discovered that one of the six wheels showed a clear bias, in that nine of the numbers (7, 8, 9, 17, 18, 19, 22, 28 and 29) occurred more frequently than the others. 

He therefore placed his first bets on 7 July 1875 and quickly won a considerable amount of money, £14,000 (equivalent to around 50 times that amount in 2005, or £700,000, adjusted for inflation). 

Over the next three days, Jagger amassed £60,000 in earnings with other gamblers in tow emulating his bets. 

In response the casino rearranged the wheels, which threw Jagger into confusion. After a losing streak, Jagger finally recalled that a scratch he noted on the biased wheel wasn't present. 

Looking for this telltale mark, Jagger was able to locate his preferred wheel and resumed winning. Counter-attacking again, the casino moved the frets, metal dividers between numbers, around daily. 

Over the next two days Jagger lost and gave up, but he took his remaining earnings, two million francs, then about £65,000 (around £3,250,000 in 2005), and left Monte Carlo never to return.

Jagger resigned from his job at the mill and invested his money in property. He is buried at Bethel Chapel, Halifax Road, Shelf.