Gambler’s Fallacy

Original article was published on Artificial Intelligence on Medium

The safest way to double your money is to fold it over and put it in your pocket.” — Kin Hubbard

Index

0. Dear Statistics Journal
1. Introduction to Probability
2. The Monty Hall Problem
3.
Gambler’s Fallacy

Let’s start, as most successful essays do, by asking a simple question.
Say I toss a fair coin 7 times and it leads to the following series of outcome:

Head, Head, Head, Head, Head, Head, Head

Now, if you had to bet on the next outcome, would you bet on it being a Heads or a Tails?

If you said Tails, well it sure seems like it should be Tails, because balance of probability! That every consecutive Head we flip incurs a debt to Tails and surely it must be much more likely for the next flip to be Tails.

Or perhaps, you’re a maverick, and you thought that the consensus would surely be Tails, but the fact that 7 heads showed up makes an even stronger case for the next flip to be Heads. *

Here’s the deal, it doesn’t matter what you picked. If you construed a pattern from the above series of flips and constructed an argument based on it to infer the next outcome, then you have committed to the Gambler’s Fallacy. In simple terms, the probability of the next outcome being a Heads or a Tails is still (50–50) and the fact that 7 consecutive heads showed up, doesn’t enforce any conditionality for the next outcome to be a Tails (or a Heads).

This is the case for all independent events.

📖 In probability, two events are independent if the incidence of one event does not affect the probability of the other event. If the incidence of one event does affect the probability of the other event, then the events are dependent.
🎰 Perhaps the most famous example of the Gambler's Fallacy occurred in a game of roulette at the Monte Carlo Casino on August 18, 1913, when the ball fell in black 26 times in a row. This was an extremely uncommon occurrence: the probability of a sequence of either red or black occurring 26 times in a row is around 1 in 66.6 million, assuming the mechanism is unbiased. Gamblers lost millions of francs betting against black, reasoning incorrectly that the streak was causing an imbalance in the randomness of the wheel, and that it had to be followed by a long streak of red.

This is the same line of reasoning that turns one bearish in a long persisting bullish market (and vice versa) as it seems obvious for processes to revert to their mean after an extremely unlikely and one arrowed tail of events.

TL;DR

The gambler’s fallacy can be attributed to the mistaken belief that gambling, or even chance itself, is a fair process that can correct itself in the event of streaks. Coincidences are bound, and in the short term, the human tendency to “find a pattern” trumps the simple nature of the fundamentals of an event. So next time you’re out gambling and you observe an unlikely event, do take a step back and then one more and then get out of there and instead spend your money on something with a higher ROI — like a lottery ticket 🙂.

Quality References

*This type of interpretation is linked to another fallacy. The Hot Hand Fallacy is the presumed phenomenon that a person who experiences a successful outcome has a greater chance of success in further attempts.

️◀ ️Previously — The Monty Hall Problem

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