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Gambler’s fallacy

Gamblers fallacy


I’ve just tossed a coin 10 times and it’s landed on heads on every toss. The next toss is almost certainly going to be tails, right!? Wrong. This is the gambler’s fallacy. A coin has no memory of what it’s landed on previously, it doesn’t want to even up the score. It is random, and because of this, whether it’s heads or tails next will always be 50-50.

What is gambler’s fallacy?

The gambler’s fallacy is the assumption that probability changes depending on past results. This change is assumed to be in the form of correcting the probability over the short term.

Our brains are great at finding patterns. Which is great in many situations, but not so when it comes to truly random events. Our minds ability to find patterns where there are none happens in every aspect of our lives. Although, as the name suggests, this fallacy is most commonly associated with the world of gambling.

Reverse gambler’s fallacy.

Very much like the standard gambler’s fallacy, the reverse version wrongly assumes that probability will change based on past results. However, this time, the gambler believes the next run of results will continue to go against correcting the probability.

An example would be, if red landed five times in a row on a roulette table, a punter that was susceptible to a reverse gambler’s fallacy mindset would assume red was more like to be the next result. As Reds are on a roll. Conversely, the typical gambler’s fallacy brain would be betting on black, thinking its blacks turn to arrive.

Of course, both these ways of thinking are wrong. Both outcomes are as likely as each other, and the casino will continually chip away at everyone betting stack with the slight edge they have inbuilt.

Gambler’s fallacy – confusing long term and short term

The confusion of what is expected in a tiny sample to that of a large sample is at the heart of what leads gamblers astray. A rather smart man, Jakob Bernoulli, came up with the Law of large number numbers many moons ago. This law states the larger your sample size of a random event, the closer you approach to the theoretical average.

Everyone understands that. And it’s why we are all happy to say a coin toss has a 50% chance of landing on either side (the theoretical average). What most people don’t take into account is the standard deviation of the short term. The table below demonstrates the difference, using the law of large numbers and standard deviation, over several different sample sizes.

No. of coin tossesStandard deviation% difference
101.5815.8
10055
100015.811.58
10000500.5
100000158.110.158
10000005000.05

As you can see, the greater the number of coin tosses, the less the end result deviates from what is expected (smaller percentage difference). But for small sample sizes, it is normal and expected, to have wild swings from the theoretically expected values.

Gambler’s fallacy is being challenged

Recently, there has been a bit of confusion surrounding the gambler’s fallacy – a mathematical principle that has been around for hundreds of years (going by the name of “the law of large numbers”). And it’s because of the many names attributed to this fallacy that the confusion has arisen.

Gambler’s fallacy has often gone by the names, these are just the names I know about, I’m sure there are many more. Monte Carlo fallacy, the fallacy of the maturity of chances, Law of Small Numbers, Alternation Bias, Negative Recency Bias, Hot Hand Effect, Sequential Decision Making, Sequential Data, Selection Bias, Finite Sample Bias, Small Sample Bias and Hot Hand Fallacy. And it’s this last name, hot hand fallacy, which has caused the confusion.

Researchers at Stanford, Berkeley, and Harvard are now arguing that the hot hand fallacy might not be a fallacy after all. But unlike using regular games of chance for their research they have been using sports, the area most readers of this blog will be concerned with. They used basketball and baseball for their research and the maths behind it is also factually correct.

They argue, “the idea that a player whose shooting percentage is higher than normal is likely to keep shooting better than normal — at least for a while” is, in fact, correct. For the purpose of sports betting, this means there may be value betting on a player or team on a hot streak. Whereas before, researchers have proven this type of winning streak to just be statistical noise.

Using an example more attuned to the readers of Jolly Odds, we’ll show an example with football. Jamie Vardy is a perfect example of the hot hand, or gambler’s fallacy, working in sports betting. For the 2015/16 season, he broke the Premier League goal scoring record for consecutive games. Scoring in 12 games in a row, and after that returning to his normal average scoring percentages.

Given this new research, it might be time to separate using all these terms to mean the same thing. Or to coin a new term for hot hands when it involves sports teams. As it no longer looks like mere statistical noise. The reasons could be with sports players’ confidence plays a massive part, or that opposition players become a little nervier. It could be that teammates now look to pass to you more often, knowing you’re on a roll. Or it could be, as was the case with Jamie Vardy, that the team now gives you more responsibility. He wasn’t the teams’ penalty taker, but during his attempt to break the record he was given penalty duties, helping skew the data.

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