- What is expected deviation in betting?
- Learn how to apply the law of large numbers to betting
- The nine-toss example
The Gambler's Fallacy
The 17th-century mathematician Jacob Bernoulli introduced the Law of Large Numbers, illustrating how a significant sample size of an event, such as flipping a coin, more accurately reflects its true probability. Despite this knowledge being around for four centuries, many bettors fall prey to what is known as the Gambler's Fallacy. This article explores why this misconception is particularly detrimental.
The Law of Large Numbers
Take the simple act of flipping a fair coin, where the probability of landing on either heads or tails is equally 50%. Bernoulli demonstrated that as you increase the number of flips, the proportion of heads to tails outcomes will converge on 50%, even though the absolute difference in the count of heads versus tails will grow.
"As the number of flips increases, the ratio of heads to tails will stabilize at 50%" This aspect of Bernoulli's theorem confounds many, giving rise to the term “Gambler’s Fallacy”. People often mistakenly predict that after a series of flips landing on heads, a tail is due next. However, since a coin lacks memory, the odds remain the same at each flip: a 50% chance for either outcome.
Bernoulli’s findings reveal that in a vast series of fair coin flips—say, a million—the distribution of heads or tails will approximate 50%. Yet, with such a large sample, an expected deviation of about 500 from an equal split is normal.
This formula helps us gauge the expected standard deviation:
0.5 × √ (1,000,000) = 500
For a large number of flips, this deviation is to be expected. However, in shorter sequences, like nine flips, this principle does not hold since the sample is too small to achieve the expected distribution, leading to potential sequences by sheer chance.
Applying Distribution in Betting
Understanding expected deviation has practical betting implications. It is particularly relevant in casino games like Roulette, where the misconception that sequences of outcomes (red or black, odd or even) will balance out within a session can lead to losses. This misconception is why the Gambler’s Fallacy is sometimes referred to as the Monte Carlo fallacy.
A notorious instance occurred in 1913 at a Monte Carlo casino roulette table, where black appeared 26 times consecutively. After the 15th black, players increasingly bet on red, wrongly assuming the probability of another black was diminishing, a classic example of the Gambler’s Fallacy.
"In 1913, a roulette table in a Monte Carlo casino saw black come up 26 times in a row. Thus, the Gambler’s Fallacy is also termed the Monte Carlo fallacy" Slot machines, essentially random number generators with a fixed Return to Player (RTP), provide another illustration. Players who have lost significant amounts may deter others from using the same machine, mistakenly believing a win is due after their losses.
For this belief to hold, one would need to engage in a vastly impractical number of plays to align with the RTP.
Jacob Bernoulli once posited that even the simplest mind could grasp that a larger sample size more accurately represents an event's true probability. Although his comment might seem a bit harsh, understanding the Law of Large Numbers—and dismissing the flawed average concept—spares one from being dubbed as one of Bernoulli’s 'simple-minded'.
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