The Gambler’s Fallacy vs Regression to the Mean - Can a Bet be "Due to Win"

In statistics, betting, and life in general, we are often faced with the dilemma of what seem to be unusual streaks of bad or good luck. When these occur, it is natural to wonder when the streak will end, whether it will be soon or whether the wait will be interminable. It is natual to wonder, "Am I due for a change in luck?" From there, it is an easy transition to decide "I must be due".

Here is a story. This happened to me recently while betting the thoroughbreds at Santa Anita racetrack. To be precise, it was the horses that were at that California racetrack. I was at a local off-track racing venue of my hometown. But the illusion of ‘being there’ can be strong, especially when caught in the moment of a lucky or unlucky streak.

In my case it was the latter. I had bet 7 straight races, using a handicapping method that had worked surprisingly well during the previous two dozen forays to the track. It involved some fairly rigorous mathematical techniques and was well-grounded in the realities of horseracing (a speed handicapping method of my own invention). In all 7 races my horse had lost, with only a couple even being vaguely close. This was highly unusual, given the method’s previous success rate.

With the final race on deck, I had to dip into some cash reserves, which I hadn’t had to do for quite some time, as I had been playing with previous winnings up until then. I was tempted to skip the final race, but then I decided against it. You might just as well finish the job, I thought. I walked up to the betting machine to purchase a win ticket for the final race, with the grim forboding of a person stuck in a losing streak.

Then the thought occurred to me – "I must be due, there is no way this losing streak can continue. It seems so obvious. Plus, I have just bought a fresh betting voucher and have more than adequate financial resources anyway. So, why not step up the bet and win it all back?"

This, I proceeded to do.

The race was confusing. The colours that the horses were wearing were difficult to make out in the stretch drive. In fact, I made the notation in a small notebook: "Lost. Also ran. Back of field." So, it looked like I wasn’t due, after all.

But then the fellow next to me said "It looks like the 6 won."

"What’s that?" I thought. "That can’t be – I bet on the 6!"

But he was right. I had mistaken the colours of the 2 horse and the 6 horse. So I was due! Not only that, I had stepped up my bet by enough to cover all my previous losses, with some to spare.

So, was I due or was I not due? Or does that even make sense?

The Gambler’s Fallacy

In an introductory statistics course, you are likely to be introduced to the notion of the gambler’s fallacy. An example of this is coin flipping. If you flip a coin a fairly large number of times, and heads comes up more often than tails, it seems intuitively obvious that you can safely bet on tails and win the game (at least until things equal out).

Unfortunately, that is not a good strategy, as the coin doesn’t have a memory – it doesn’t know that tails are due. In the long run, the percentage of heads vs tails will tend to 50:50, but that can take a long time. Indeed, the preponderance of heads can continue to grow in raw numbers, even as the percentage of heads drops and approaches 50%. This does seem rather counter-intuitive, but it can be proved mathematically via the use of something called the mean value theorem.

Regression to the Mean

This is a different but related concept. Basically, it says that in some complex process (more complicated than coin-flipping), if a participant begins to perform much better (or worse) than its long-term average, the participant will regress back towards that average (fairly soon). So, for example, a hockey player on a heater (e.g. scoring streak) will likely go back to his long-term scoring, given enough time. Same for a horse (or some category of horse) at the race track. It is also works in genetics, where offspring tend to revert back to their ancestors in some characteristic, short of selective breeding.

How is this different from coin-tossing. I would venture to say that it is different because these processes do have a memory. Hockey players who go on a heater will generally be having some puck-luck, which can create as a high confidence level, at least temporarily. Opponents will become aware of the player’s scoring streak and actively try to subvert it (that’s what the game is all about). He will be shadowed by the opposing team’s best checker. He may be singled out for some rough play. The player’s confididence may then drop, as will his performance.

Similarly for horseraces. The horse has a memory, though it is difficult to say how much that affects a horse. However, trainers and jockeys definitely have memories that will affect the process. A hot horse may get better treatment from trainers and jockeys, thus helping its win streak. On the other hand, it may be run too often and get physically worn out. Its betting odds will drop, which may motivate trainers and jockeys to darken its form, so as to improve its odds. Horses running against it will have better odds, which may ensure that jockeys and trainers go to extra lengths to beat the hot horse.

So, basically, there are multiple participants in these types of complex games, all with their own goals and strategies (thousands to millions in the stock market). The combination of these actor’s strategies will tend to cancel each other out, thus tending to bring all the opponens back to long-term performance metrics.

Conclusion

So, which phenomenon prevails? The Gambler’s Fallacy (don’t base actions on the notion of something being ‘due’) or Regression to the Mean (you probably can base actions on something being due).

I don’t really know. I am conflicted. But I think it is probably a reasonable strategy for complex games (sports, stock markets, predictive markets) but not for relatively simple games.

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And here are a couple of short works that you might enjoy, if these matters interest you.

A Dark Horse

In “A Dark Horse”, a gambler’s desire to hit a big win seems to lead him to make a Faustian bargain with a supernatural evil. Or is it all just a string of unnaturally good luck?

The story is just $0.99 U.S. (equivalent in other currencies) and about 8000 words. It is also available on Kindle Unlimited and is occasionally on free promotion.

U.S.: https://www.amazon.com/dp/B01M9BS3Y5

U.K.: https://www.amazon.co.uk/dp/B01M9BS3Y5

Germany: https://www.amazon.de/dp/B01M9BS3Y5

France: https://www.amazon.fr/dp/B01M9BS3Y5

Italy: https://www.amazon.it/dp/B01M9BS3Y5

Netherlands: https://www.amazon.nl/dp/B01M9BS3Y5

Spain: https://www.amazon.es/dp/B01M9BS3Y5

Japan: https://www.amazon.co.jp/dp/B01M9BS3Y5

India: https://www.amazon.in/dp/B01M9BS3Y5

Mexico: https://www.amazon.com.mx/dp/B01M9BS3Y5

Brazil: https://www.amazon.com.br/dp/B01M9BS3Y5

Canada: https://www.amazon.ca/dp/B01MDMY2BR

Australia: https://www.amazon.com.au/dp/B01M9BS3Y5

Here’s an interesting review from Goodreads

(BTW, the writer has never met the reviewer and was not even aware of the review until very recently – You can look up the review on Goodreads, if you like)

A Dark Horse

Every gambler is bound to run out of luck eventually, right?

By far my favourite type of horror is psychological horror. I was quite pleased with how Mr. Olausen frightened his audience without spilling a single drop of blood or so much as hinting at anything gory. He knew exactly what hints to drop for us that made us deliciously dread the next scene simply by throwing out hints about who or what the dark horse might actually represent. This is the kind of stuff I love getting scared by, especially as Halloween approaches.

It would have been helpful to have more character development in this short story. While I certainly wouldn’t expect to see as much time spent on this as I would for a full-length novel, I did have trouble connecting to the main characters due to how little I knew about them and how much their personalities seemed to remain the same no matter what happened to them. If not for this issue, I would have felt comfortable choosing a much higher rating as the plot itself was well done.

I must admit to not knowing much about gambling at all, so I appreciated the brief explanations the narrator shared about how placing bets works and why some people have so much trouble walking away from a bet. While I will leave it up to experts on these topics to say how accurate everything was, I did enjoy learning more about the main character’s addiction and what he hoped to gain from betting on just one more game or race. It gave me a stronger sense of empathy for folks in his position.

A Dark Horse – A Gothic Tale was a deliciously chilling story for the Halloween season and beyond.

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And here is an account of a summer job on the railroad, during my undergraduate years.

One Summer Working on the Railroad

What follows is anaccount of a few weeks one long-ago summer, when I was 19 and was working for the Canadian National Railway (CNR) on a railroad construction gang, in the wilds of north-central British Columbia, Canada.

The journal is in the form of a letter, that was never sent. Decades later, I think it has an interesting historical resonance. At times I come off like a callow youth – I plead guilty as charged. I swore a lot more in those days than I do now, but in places the writing is surprisingly good, at least in my humble opinion. And the story has a compelling narrative arc.

There were a lot of interesting and dramatic events that occurred – a number of industrial accidents being the most serious. There were also some colorful characters on the crew, which resulted in some dramatic and at times amusing conflicts and altercations. I perhaps flatter myself by including myself in that number. Or perhaps I condemn myself – I’m not sure.

So, if you want to be reminded of one of those summer jobs that was kind of life-changing, read on. My story may just kick-start some memories of your own.

The memoir/journal is about 9,000 words, a length that can usually be read in an hour or so. It is priced at 99 cents U.S. (equivalent in other currencies) and is free on Kindle Unlimited. Periodically, it will be offered as a free promotion.

U.S.: https://www.amazon.com/dp/B0CN661P8Z

UK: https://www.amazon.co.uk/dp/B0CN661P8Z

Canada: https://www.amazon.ca/dp/B0CN661P8Z

Australia: https://www.amazon.com.au/dp/B0CN661P8Z

India: https://www.amazon.in/dp/B0CN661P8Z

Germany: https://www.amazon.de/dp/B0CN661P8Z

France: https://www.amazon.fr/dp/B0CN661P8Z

Spain: https://www.amazon.es/dp/B0CN661P8Z

Italy: https://www.amazon.it/dp/B0CN661P8Z

Netherlands: https://www.amazon.nl/dp/B0CN661P8Z

Japan: https://www.amazon.co.jp/dp/B0CN661P8Z

Brazil: https://www.amazon.com.br/dp/B0CN661P8Z