Monte Carlo Roulette 1913
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 most famous example came in a game of roulette in a Monte Carlo casino in 1913 (hence the name) when the ball fell in black 26 times in a row. Gamblers lost millions betting against black. Monte Carlo decreases with the inverse of particle his- tories, particle splitting and Russian roulette tech- niques are used in TRIPOS. The importance zone is taken to be the first few atomic layers where most of the sputtered particles orginate. The analog Monte Carlo code TRIPOS is.
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.
'As the number of tosses get larger the distribution of heads or tails evens out to 50%'
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).
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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:
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.
'In 1913, a roulette table in a Monte Carlo casino saw black come up 26 times in a row. For that reason, gambler’s fallacy is also known as Monte Carlo fallacy'
Monte Carlo Roulette 1913 Youtube
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, but 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’.
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The term Gambler's fallacy refers to a misconception about statistics. It is also known Monte Carlo fallacy or fallacy of the maturity of chances. In statistics, a random event has a certain probability of occurring. The fallacy is that if the event has occurred more frequently in the past, it will occur less frequently in the future; or that if it has been less frequent in the past, it will be more frequent in the future.
Childbirth[change change source]
As early as 1796, the idea of the gambler's fallacy was used to 'predict' the sex of children. In his work A Philosophical Essay on Probabilities, published in 1796, Pierre-Simon Laplace wrote of the ways in which men calculated their probability of having sons:'I have seen men [who wanted to have] a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers. Imagining that the ratio of these births to those of girls ought to be the same at the end of each month, they judged that the boys already born would render more probable the births next of girls.' In short, the expectant fathers feared that if more sons were born in the surrounding community, then they themselves would be more likely to have a daughter.[1]
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Some expectant parents believe that, after having multiple children of the same sex, they are 'due' to have a child of the opposite sex. While the Trivers–Willard hypothesis predicts that birth sex is dependent on living conditions (i.e. more male children are born in 'good' living conditions, while more female children are born in poorer living conditions), the probability of having a child of either sex is still generally regarded as near 50%.
References[change change source]
- ↑Barron, Greg; Leider, Stephen (2010). 'The role of experience in the Gambler's Fallacy'. Journal of Behavioral Decision Making. 23 (1): 117–129. doi:10.1002/bdm.676. ISSN0894-3257.