Authored by Ion Saliu (himself!) on November 18, 2011; last update December 2, 2011.
The infamous Gambler's Fallacy was the first bullet shot at yours truly, as it were. I was promoting the Fundamental Formula of Gambling (FFG) and potentially winning gambling systems derived from the FFG. My opponents, most of them vociferous and aggressive, posed me with this “mathematical” situation:
“You lose once; the probability that you will lose next time is the same as before. You lose twice; the probability that you will lose the third time in a row is the same as before.... and so on.... ad infinitum, if you will....” It's like a god of odds will always “create” endless losing outcomes for the player (but always winning situations for the gambling house)!
I've had skirmishes with many a gambling author or casino executive. Chief among them was one “Wizard of Odds” (self-proclaimed after the name of a TV show!) The “Wizard” stated in an email to me that getting 200 consecutive heads in coin tossing has the same chance as getting just one heads!
The concept of degree of certainty was unknown - terra incognita until I published the Fundamental Formula of Gambling in 1997 — zero results in all search engines. Google and Wikipedia not even born then! Probability was the only concept in the argument. Indeed, the probability of getting heads in coin tossing is always the same (1/2): In the 1st toss, the 2nd throw, the 3rd throw.... the Nth toss.... But the degree of certainty is different from toss to toss and it is dependent on the third fundamental parameter of probability, number of trials. The degree of certainty, DC, increases with the increase in the number of trials, N, while the probability, p, is always constant.
My forums, new and old, always have been a place of debating the gambler's fallacy. A usually well intended visitor posed me with this situation:
“Regarding the 'gambler's fallacy', I just wanted to highlight that one can't be sure that the strategy that works in that particular case will work again.”
That was a case of... double gambler's fallacy! Even if the player wins one time, the win will never repeat!
I made a strong case against the gambler's fallacy and a logically necessary reversed gambler's fallacy:
Wizard of Odds Had High Praise for Ion Saliu's Gambling Theory.
Gambler's Fallacy, Reversed Gamblers Fallacy, Probability, Streaks.
There are losing streaks and there are winning streaks — that's mathematics. The casinos and many gambling authors, however, grossly disregard probability theory. They established a "law" that "assures" very long losing streaks for the player — but never for the gambling establishment. Don't give me that bull — the gambler's fallacy! If there is honesty, things should closely resemble the calculations of my probability software Streaks.EXE. There should be also a reversed gambler's fallacy, Lady Chance says — but Wizard of Odds curses when he hears that!
That program, Streaks.EXE, caused fundamental changes in probability theory. It proved the absurdity of the concept of gambler's fallacy by presenting another absurdity: the reversed gambler's fallacy.
Wikipedia edited its pages on the gambler's fallacy, also adding the reversed gambler's fallacy. They never mention my name, let alone give me credit. Many people complain about those lowly-character practices of Wikipedia. Among other things, they took also my concept of Degree of Certainty and … refer to others! (That Wikipedia page is listed at position #1 by search engines … although it has no content … only 3 links!) Not only that I coined the concepts, but I am still the only one who wrote the software to calculate and disprove or prove the concepts!
One writes about a new and important topic — Wikipedia will write an article soon after they visited your Web site! The same happened with the Reversed Birthday Paradox, or Monty Hall Paradox — to name just a few cases I discovered recently. The Wikipedia updates follow directly updates to my Web pages and software. But they still don't know things I haven't published, such as the formulas or algorithms in my software! Granted, I've always been critical of Wikipedia and keep them in low regards — revenge is sweet, they say.... Or, they might be afraid that crediting me would raise the immediate suspicion of ... blunt inspiration!
No wonder Wikipedia so vehemently opposes a bill introduced in the House Judiciary Committee of the U.S. Congress (2011)! The bill is known as Stop Online Piracy Act (SOPA). Simply, Wikipedia cannot live without piracy. Why don't they, Wikipedians, just give proper credit to the legitimate authors?!
Lately, several gambling forums talk about the two absurdities — but no longer mention my name! But the main force behind the demise of the gambler's fallacy is the Fundamental Formula of Gambling (FFG). Also, my theory and, systems and especially software contributed big time. My probability software (bundled in a huge collection of programs named Scientia.EXE) offers the most accurate verification instruments. Even my totally free online applications are great probability tools: Probability calculator, random number generator.
I ran my random generator for roulette many, many times. I have never seen situations even remotely close to the “predicaments” of the gambler's fallacy or the reversed gambler's fallacy. I chose the roulette because the odds are more manageable, so the number of trials will satisfy the law of large numbers by a wide margin. The roulette module generates 1000 spins at a time. Probability for a single number in double-zero roulette is p = 1/38. The number of trials 1000 is 26 times larger than N (as in p = 1/N). All my probability calculations prove that a number of trials larger than N * 20 fully satisfy the law of large numbers.
Furthermore, I ran the roulette function 20 times. I recorded only run #20. I ran again the roulette function 20 times. Again, I recorded only the last run. I scroll down to the bottom of the ActiveX control.
I look at the two-dozen bets. They are components of my Super Roulette Strategy. Of course, we will not see a 2-dozen combination with 0 appearances, or one 2-dozen combination with 1000 wins! I've never seen that — and no sane person will see such an occurrence in one or two lifetimes! There is always at least one winning two-dozen combination AND at least one losing two-dozen combination.
I calculated the binomial standard deviation (BDF) with SuperFormula.EXE: function D = Standard Deviation. It calculates the binomial NOT the statistical standard deviation. The statistical form of deviation is calculated in function S = Miscellanea, then function 2 = Stats of Data.
The probability for one two-dozen group is 24/38. The binomial standard deviation for a two-dozen combination in 1000 roulette spins is 15.25 (closer to 15 than 16 spins). We can notice in a vast majority of cases that most deviations from the norm (expected number of successes) fall within 1 standard deviation. The normal probability rule is very lax, therefore not nearly as precise as the Fundamental Formula of Gambling (FFG). The margin of generosity in the normal probability rule is way too large. (Just you wait for a new article in Wikipedia on margin of generosity in statistics! Hey, I even got a formula for that!)
The casinos would want people believe that there is no formula for calculating the deviation from the norm or the number of streaks. It's normal, they would try to convince you, to lose 10 times in a row in 15 blackjack hands; then, lose 15 times in a row in the next 20 hands. But, then again, don't ask them why the house doesn't lose the same way, especially when the bet is higher! “The god of odds is on our side,” they might answer.
The gambler's fallacy and the reversed gambler's fallacy come closest to life in two incredible situations. They occur in online gambling. It has been noticed by many players, including myself. The two situations are: playing for fun and playing with real money. The first situation grants the player with very long winning streaks, and big wins (the reversed side of the fallacy). The second situation hurts the player with very long losing streaks, especially when the bet is higher or maximum; the real gambler's fallacy!
Those who have serious arguments (read, fights) with me, always produce “real” casino situations when they encounter extremely long losing streaks — while applying my gambling systems! Yet, they never show to me cases of … reversed gambler's fallacy — long winning streaks!! It is all about winning and losing, in streaks mathematically calculated. Also, I never advise open-ended martingales — I always advise limited-step Martingale betting. Also, the longer the gambling session, the higher the degree of certainty to encounter longer streaks. There will be a larger number of long losing streaks — but there will be long winning streaks as well. Just keep track of all streaks and martingale accordingly …