About Gambler’s Fallacy
A detailed overview of the Gambler’s Fallacy of the Law of Large Numbers
Established way back in the 17th century by someone known as Jacob Bernoulli, the Law of Large Numbers shows or proves the fact that the larger the sample of a certain event is, for instance a coin toss, the more accurately you’ll be able to represent its accurate probability. Sports bettors or gamblers in general have been struggling with this idea for around 400 years now, a major reason why it has started being referred to as the Gambler’s Fallacy. Let’s discover why this mistake can prove to be very expensive for sports bettors or any gambler in general.
What exactly is the Law of Large Numbers?
If we take a look at the example of a fair coin toss, wherein your chance of getting tails or heads is an equal 50%, Jacob Bernoulli had calculated that with the number of coin tosses becoming larger, the exact percentage of tails or heads results becomes closer and closer to the 50% mark. At the same time, the difference between the exact number of tails or heads (gotten from coin toss) also becomes larger.
This is fairly easy to understand, but it’s the second part of Bernoulli’s theory which people face great difficulty understanding – which has also resulted in it getting termed as the Gambler’s Fallacy. Tell someone that you’ve flipped a coin nine times and have gotten heads in each one of those instances, and that person’s prediction for the next coin toss is most likely to be tails.
It is actually incorrect, simply because a coin doesn’t have any memory and hence every time you toss it up, the probability of getting tails or heads remains the same – which is 0.5 or 50%.
Jacob Bernoulli’s discovery or theory revealed that with the sample of fair coin tosses getting bigger, for instance, reaching in the vicinity of 1 million, the distribution of tails or heads would also even out and close in on the figure of 50%. Owing to the huge size of the sample, however, expected deviations from an accurate 50 – 50 split can possibly be as high as 500.
The following equation which calculates the statistical standard deviation provides us with a fair idea of what we should be expecting:
0.5 x √ (1,000,000) = 500
Although you can easily observe the expected deviation when it comes to these many number of coin tosses, in case of the nine toss example we referred to earlier, the sample is just not large enough for this calculation to apply.
Hence, the nine tosses can be termed as a small extract from that huge 1 million tosses; and the sample is actually too tiny to even out, as suggested by Jacob Bernoulli over a sample as large as of 1 million tosses. It can instead create a sequence of its own by pure chance.
Application of distribution in sports betting
If you carefully look into the sports betting theory you will observe that there are some very clear applications for the expected deviation in it. Its most obvious application can be seen in casino games such as roulette, wherein a gambler’s misplaced belief that the sequences of even or odd, or black or red, are most likely to even out in any given session of play. This belief can often leave a gambler out of pocket in no time! That’s also the precise reason that Monte Carlo fallacy is the other popular name given to the Gambler’s Fallacy.
A roulette table in a Monte Carlo casino witnessed black coming up 26 times in a row back in the year 1913. Once the number of Blacks in a row reached 15th instance, the gamblers were all piling onto red, strongly believing that the chances of getting one more black was almost astronomical, thereby showing an irrational belief among gamblers that the next spin of a roulette table is somehow influenced by the present one. Which we all know is not the case!
Another popular example can be seen in case of slot machines, which in effect are nothing but random number generators, with a certain preset Return to Player (RTP). You may often witness slot machine players who’ve already pumped large sums of money into some slot machine without achieving any success, stopping other slots players from playing on their machines, strongly convinced that a huge win is almost round the corner and would end their losing streak.
However, it should be noted that for this tactic to deliver any sorts of viable results, the gambler should have had played a huge number of slots, in order to reach that Return to Player or RTP.
At the time when Jacob Bernoulli had established his law, he had strongly asserted that even the most stupid man on the planet understands that bigger the sample, the higher are the chances of it representing the actual probability of a certain event. Although many believe that he was slightly harsh in making that sort of assessment, nothing can deny that once you have a good understanding of the Law of Averages and the Law of Large Numbers, you would never be counted amongst the stupid men that Bernoulli had referred to!