Mathematical Formula For Blackjack

Math maths formula formulae. God is absurd it is worse than impossible! Leads mathematical formula that god is a kokostirk. Maths formulas for everyone. Insider information to stock market is illegal and not mathematical. Chance to win gambling. Mathematical foundation of gambling, lottery, software. Certainties of hazard. Dealer Ace) = 52,992 / 15,039,960 = 0.003523. This means that even money situations come up about 35 times per 10,000 hands, or about once every 284 hands. Even money is far more common, and therefore potentially far more profitable for the casino, in a shoe game.

1. Mathematical Formula For Blackjack Games
2. Mathematical Formula For Blackjack Card Game
3. Mathematical Formula For Blackjack For Dummies
4. Mathematical Formula For Blackjack Rules

.015? Huh? What am I missing here?

.015? Huh? What am I missing here?

wondering the same thing

Blackjack - 70 hands/hr at 0.75% (for comps)

http://wizardofodds.com/gambling/house-edge/Interesting thread, thanks for starting it, mds. 'Theo' is a mystical thing hosts hint at but never discuss directly with clients; at least that's how it is where I go. I have heard the 'four hours of play a day' mentioned many times over the years. That seems to be the gold standard for defining serious customers. That is, I believe the play-time parameter in the formula is regarded as more important that it's mathematical weight when hosts evaluate a customer. I have tried to educate some folks about how they are evaluated. They whine about how much money they have lost, thinking that is the only measure of worth. My standard line is, 'They only care about how much you bet, your luck or lack of it is your problem, not theirs.' That reasoning is seldom believed.

MOST casinos base comps on your theoretical value (ie, expected losses) and only adjust this figure for massive actual losses and less often for massive actual wins. They focus on what you risked rather than how you actually did. Since your long term value to the casino is what you able and willing to RISK, not your personal variance in the short term. The casino focuses on your long term value to them and protects that value by short term actions to keep you happy.

Although the comps should based on expected loss rather than actual losses - the house does it for different reasons than stated above. If comps are based on actual losses, the house is widely open for systematic exploitation.

Interesting thread, thanks for starting it, mds. 'Theo' is a mystical thing hosts hint at but never discuss directly with clients; at least that's how it is where I go. I have heard the 'four hours of play a day' mentioned many times over the years. That seems to be the gold standard for defining serious customers. That is, I believe the play-time parameter in the formula is regarded as more important that it's mathematical weight when hosts evaluate a customer. I have tried to educate some folks about how they are evaluated. They whine about how much money they have lost, thinking that is the only measure of worth. My standard line is, 'They only care about how much you bet, your luck or lack of it is your problem, not theirs.' That reasoning is seldom believed.

I get that all the time from friends. They lose 3000 in an hour or less and want 3 nights at RFB or RFBI... Just doesn't work that way. Most importantly, to the Casino it's about hours played. As you put in more play time that's when you will lose. You will get comped more but you will lose at the tables. In other words, just pay for your room food and beverage yourself! Why lose 4000 at the tables to get a 200 a night room comped? Short term winners and long term losers. Except if you are an AP. Should be opposite.

Although the comps should based on expected loss rather than actual losses - the house does it for different reasons than stated above. If comps are based on actual losses, the house is widely open for systematic exploitation.

Actual losses if rather whopping or early in one's stay will merely 'sweeten' the comps that are based on a player's theo. The sweetener is modest since in most circumstances only Theo is considered.
Time at Table can be three hours but it seems four is simply more impressive by a profound amount and it is rumored that at the Venetian the four hour mark is critical.

• Page 1 of 2

By Ion Saliu,
Founder of Gambling Mathematics, Founder of Probability Theory of Life

I. Logic of Fundamental Formula of Gambling
II. The Mathematical Solution: Divine Logarithm
III. Mathematics of the God Concept: Formula of Absurdity

• Why God Fears Mathematics, While Einstein Hates Gambling...

IV. Insider Information: It's All in Our Reason
V. Mathematics of Ion Saliu's Paradox or Problem of N Trials
VI. Software: Divine Tool to Further Empower Reason
VII. Links and Resources Regarding Mathematics, Probability

1. Logical Steps, Algorithm to the Fundamental Formula of Gambling

First captured by the WayBack Machine (web.archive.org) on May 10, 2000.

'Let no one enter here who is ignorant of mathematics.'
(The frontispiece of Plato's Academy)

'The most important questions of life are, for the most part, really only problems of probability.'
(Pierre Simon Marquis de Laplace, 'Théorie Analytique des Probabilités')

Here is how I arrived to, by now, famous Fundamental Formula of Gambling (FFG). When laypersons say: 'It is so simple,' it always represents undeniable mathematics; therefore, undeniable Truth. I thought I had worked it out on my own, because the formula starts with the very essence of probability theory: p = n/N, or reduced to a p = 1/N mathematical relation. A truth becomes (almost) self–evident when a number of people think independently of the same thing. But this human law must be the most undeniable of them all: No Truth is self–evident AND no human thinks totally independently of others.

My first step got my feet wet in my pick 3 lottery software pond. 'Probably win,' that's what I had thought.

I rationalized in this manner. The probability of any 3-digit combination is 1/1000. Therefore, I had expected that the repeat (skip) median of a long series of pick-3 drawings would be 500. It would be similar to coin tossing, where the median of p=1/2 series is 1. In other words, the median of a long series of coin tosses is 1. To my surprise, the repeat median of long series of pick-3 drawings was not 500. It was closer to 700.

I checked it for series of 1000 real drawings and also randomly generated drawings. Then I checked the median against series of 10000 (10 thousand) drawings. The median of the skip was always close to 700. Do not confuse it for the median combination in the set. That value of the median is, in fact, either 499 or 500. The correct expression is 4,9,9 or 4 9 9 or 5,0,0 (three separate digits).

What is that median useful for, anyway? Among other properties, the skip median (or median skip) shows that, on the average, a pick-3 combination hits in a number of drawings. Any pick-3 combination hits within 692 drawings in at least 50% of the cases. Equivalently, if you play one pick-3 combination, there is a 50%+ chance it will hit within 692 drawings, or it will repeat no later than 692 drawings. The chance is also (almost) 50% that you will have to wait more than 692 drawings for your number to hit.

I studied theory of probability (gambling mathematics too!) in high school and in college. Some things get imprinted on our minds. Such information becomes part of our axioms. An axiom is a self-evident truth, a truth that does not necessitate demonstration. We operate with axioms in a manner of automatic thinking.

So, I was analyzing mathematically long pick-3 series, where p=1/1000. Next, I wrote the probability formula of a single pick-3 number to hit two consecutive drawings: p = 1/1000 x 1/1000 = 1/1000000 (1 in 1 million). I have never found useful to work with very, very small numbers in probability.

How about the reverse? The probability of a particular pick-3 number NOT to hit is p = (1 – 1/1000) = 999/1000 = 0.999. This is a very large number. It is almost certain that my pick-3 combination will not hit the very first time I play it.

How about not hitting two times in a row? P = (1 – 1/1000) ² = 0.999 to the power of 2 = 0.998. Still, a very large number! I reversed the approach one more time: What is the opposite of not hitting a number of consecutive drawings? It is winning within a number of consecutive drawings.

The knowledge was inside my head. Unconsciously, I used Socrates' dialectical method of delivering the truth. (His mother delivered babies.) I also followed steps in De Moivre formula. At this point, I had this relation:

1 – (1 – p) ^N

where N represents the number of consecutive drawings.

I thought that for N = 500 drawings, the expression above should give the median, or a probability of 50%. So, I calculated 1 – (1 – p) ^N = 1 – (1 – 1/1000) ⁵⁰⁰ = 1 – (0.999) ⁵⁰⁰ = 0.3936 = 39.36%. Thus, my relation became:

0.3936 = 1 – (1 – 1/1000) ⁵⁰⁰

I made N = 692. I obtained the value:
1 – (1 – 1/1000) ⁶⁹² = 1 - 0.5004 = 49.96% (very close to 50%).

Next step: I made N = 693. I obtained the degree of certainty:
1 – (1 – 1/1000) ⁶⁹³ = 1 - 0.4999 = 50.01% (very, very close to 50%).

Thus, the parameter I call the FFG median is between 692 and 693 for the pick-3 lottery.

2. The Mathematical Solution: Divine Logarithms

I concluded I should not make more assumptions. What if I don't think I know what N should be for the median (50%), or for any other chance, which I simply called the degree of certainty? I realized I had the liberty to select whatever degree of certainty I wanted to, and only had to calculate N. The relationship became:

DC = 1 – (1 – p) ^N

Then:

(1 – p) ^N = 1 – DC

The equation can be solved using logarithms:

The only unknown is N: the number of consecutive drawings (or trials) that an event of probability p will appear at least once with the degree of certainty DC.

The rest is history. I called the relation “The Fundamental Formula of Gambling” almost automatically. Unintentionally, it might sound cocky. Just refer to it as FFG.

Nothing comes with absolute certainty, but to a degree of certainty! That's mathematics, and that's the only TRUTH.

3. Mathematics of the God Concept: The Formula of Absurdity

Unfortunately, O glorious sons and daughters of Logos and Axioma, the idea of God is a mathematical absurdity! It saddened me first, but we must come to grip with reality. The humans fictionalize because we feel we can't live without the comfort of absolute certainty.

* Some humans with mathematical skills will stumble upon an error, when the degree of certainty DC is set to 100%. There is no absolute certainty in the Universe (or probability equal to 1). It leads to an absurdity: Calculating the number of events necessary for an event of probability p to appear with a degree of certainty equal to 100%. It is absurd. No other qualifications apply, such as impossible or erroneous formula. Just remember the relation we had before considering the degree of certainty! We dealt with the probability of losing N consecutive times: (1 – p) ^N. In this relation, no N can lead to zero (1 – 100%). Not even minus Infinity! A computer program should trap the error and ask the user to enter a DC less than 100% (things like 99.99999999…%).*

Some profound thoughts surrounding this mathematical expression and the false error.
The Fundamental Formula of Gambling (FFG) proves that absolute certainty is a mathematical absurdity. If we set the degree of certainty DC=1 (or 100%), FFG leads to a mathematical absurdity. God is Absolute Certainty, therefore absolute absurdity. I can only imagine de Moivre's reaction when this thought might have crossed his mind: 'Certainty is absurd! How can God be True?' It was the 17th century, and the 21st has just a little changed for the better…

Aporia is a special form of absurdity. A Sophist philosopher — Zeno of Elea — constructed a most famous aporia. It is known as the Paradox of Achilles and the tortoise. Read the first philosophical, logical, and mathematical solution: Zeno's Paradox: Achilles Can't Outrun the Tortoise?

It was very easy to apply FFG mathematics to many types of lottery and gambling games. It is at the foundation of several high probability gambling systems I designed: roulette, blackjack, horseracing, and lottery. It works with stocks, too. There is significant randomness in stock evolution. Many stockbrokers came to terms with the reality that all stocks fluctuate in an undeniably random fashion. I am surprised how many brokerage firms have visited my site!

4. Insider Information: It's All in Our Reason

In the year of grace 2001 my memory dug out a real gem. I wrote about it in a post on my message board: Cool stories of the Truth.

“I found another treasure: A little book in Romanian. Don't they say great things come in small packages? It couldn't be truer than in this case. The book was The Certainties of Hazard by French academician Marcel Boll. The book was first published in French in 1941. My 100-page copy was the 1978 Romanian edition. It all came to life, like awakening from a dream. The book presented a table very similar to the table on my Fundamental Gambling Formula page. Then, in small print, the footnote: “The reader who is familiar with logarithms will remark immediately that N is the result of the mathematics formula: N=log(1-pi)/log(1-p).”

That's what I call the Fundamental Formula of Gambling, indeed! Actually, the author, Marcel Boll did not want to take credit for it. Abraham de Moivre largely developed the formula. Then I remembered more clearly about de Moivre and his formula from my school years. Abraham de Moivre himself probably did not want to take credit for the formula. As a matter of fact, the relation only deals with one element: the probability of N consecutive successes (or failures). Everybody knows, that's p ^N (p raised to the power of N). It's like an axiom, a self-evident truth. Accordingly, nobody can take credit for an axiom. I thought Pascal deserves the most credit for establishing p = n / N. From there, it's easy to establish p ^N. And give birth to so many more worthy numerical relations.

5. Mathematics of Ion Saliu's Paradox or Problem of N Trials

Another look at one of the steps leading to the Fundamental Formula of Gambling:1 – DC = (1 – p)^NWe can express the probability as p = 1 / N; e.g. the probability of getting one point face when rolling a die is 1 in 6 or p = 1 / 6; the probability of getting one roulette number is 1 in 38 or p = 1 / 38. It is common sense that if we repeat the event N times we expect one success. That might be true for an extraordinarily large number of trials. If we repeat the event N times, we are NOT guaranteed to win. If we play roulette 38 consecutive spins, the chance to win is significantly less than 1!1 – DC = (1 – 1 / N) ^N

I noticed a mathematical limit. I saw clearly: lim((1 – (1 / N)) ^N) is equal to 1 / e (e represents the base of the natural logarithm or approximately 2.71828182845904...). Therefore:

1 – DC = 1 / eandDC = 1 — (1 / e)

The limit of 1 — (1 / e) is equal to approximately 0.632120558828558...

I tested for N = 100,000,000... N = 500,000,000 ... N = 1,000,000,000 (one billion) trials. The results ever so slightly decrease, approaching the limit … but never surpassing the limit!

Mathematical Formula For Blackjack Games

When N = 100,000,000, then DC = .632120560667764...
When N = 1,000,000,000, then DC = .63212055901829...

(Calculations performed by SuperFormula, function C = Degree of Certainty (DC), then option 1 = Degree of Certainty (DC), then option 2 = The program calculates p.)

If the probability is p = 1 / N and we repeat the event N times, the degree of certainty DC is 1 — (1 / e), when N tends to infinity. I named this relation Ion Saliu's Paradox of N Trials.

Soon after I published my page on Theory of Probability, Ion Saliu Paradox (in 2004), the adverse reactions were instantaneous. I even received multiple hostile emails from the same individual! Basically, they considered my (1 / e) discovery as idiocy! “You are mathematically challenged”, they were cursing! Guess, what? I saw in 2012 an edited page of Wikipedia (e constant) where my (1 / e) discovery is considered correct mathematics. Of course, they do not give me credit for that. Nor do they demonstrate mathematically the (1 / e) relation — because they don't know the demonstration (as of March 21, 2012)!

• You can see the mathematical proof right here, for the first time. I created a PDF file with nicely formatted equations:

• Mathematics of Ion Saliu Paradox.

How long is in the long run? Or, how big is the law of BIG numbers? Ion Saliu's paradox of N trials makes it easy and clear. Let's repeat the number of trials in M multiples of N; e.g. play one roulette number in two series of 38 numbers each. The formula becomes:

1 – DC = (1 – 1 / N)^NM = {(1 – 1 / N)^N}^M = (1 / e)^M

Therefore, the degree of certainty becomes:

DC = 1 – (1 / e)^M

Mathematical Formula For Blackjack Card Game

If M tends to infinity, (1 / e)^M tends to zero, therefore the degree of certainty tends to 1 (certainty, yes, but not in a philosophical sense!)

Actually, relatively low values of M make the degree of certainty very, very nearly 100%. For example, if M = 20, DC = 99.9999992%. If M = 50, the PCs of the day calculate DC = 100%. Of course, they can't approximate more than 18 decimal positions! Let's say we want to know how long it will take for all pick-3 lottery combinations to be drawn. The computers say that all 1000 pick-3 sets will come out within 50,000 drawings with a degree of certainty virtually equal to 100%.

Ion Saliu's Paradox of N Trials refers to randomly generating one element at a time from a set of N elements. There is a set of N distinct elements (e.g. lotto numbered-balls from 1 to 49). We randomly generate (or draw) 1 element at a time for a total of N drawings (number of trials). The result will be around 63% unique elements and around 37% duplicates (more precisely named repeats).

Let's look at the probability situation from a different angle. What is the probability to randomly generate N elements at a time and ALL N elements be unique?

Let's say we have 6 dice (since a die has 6 faces or elements); we throw all 6 dice at the same time (a perfectly random draw). What is the probability that all 6 faces will be unique (i.e. from 1 to 6 in any order)? Total possible cases is calculated by the Saliusian sets (or exponents): 6⁶ (6 to the power of 6) or 46656. Total number of favorable cases is represented by permutations. The permutations are calculated by the factorial: 6! = 720. We calculate the probability of 6 unique point-faces on all 6 dice by dividing permutations to exponents: 720 / 46656 = 1 in 64.8.

We can generalize to N elements randomly drawn N at a time. The probability of all N elements be unique is equal to permutations over exponents. A precise formula reads:

Probability of unique N elements = N! / N^N

I created another type of probability software that randomly generates unique numbers from N elements. I even offer the source code (totally free), plus the algorithms of random number generation (algorithm #2). For most situations, only the computer software can generate N random elements from a set of N distinct items. Also, only the software can generate Ion Saliu's sets (exponents) when N is larger than even 5. Caveat: today's computers are not capable of handling very large Saliusian sets!

I wrote also software to simulate Ion Saliu's Paradox of N Trials:
OccupancySaliuParadox, mathematical software also calculating the Classical Occupancy Problem.

Read more on my Web page: Theory of Probability: Introduction, Formulae, Software, Algorithms.

6. Software: The Divine Tool to Further Empower Reason

I wrote software to handle the Fundamental Formula of Gambling (FFG) and its reverse: Anti-FFG or the

Mathematical Formula For Blackjack For Dummies

Degree of Certainty. There are situations when we want to calculate the Degree of Certainty that an event of probability p will appear at least once within a number of trials N. As a matter of fact, this method offers a more precise correlation between an

Mathematical Formula For Blackjack Rules

integer number of trials and a degree of certainty DC expressed as a floating-point number. Furthermore, the program can determine the probability from a data series! The number of elements in the data series is known (N). Sorting the data series can determine the median: The degree of certainty DC equal to 50%!

FORMULA calculates several mathematical, probability, and statistics functions: Binomial distribution; standard deviation; hypergeometric distribution; odds (probability) for lotto, lottery, and gambling; normal probability rule; sums and mean average; randomization (shuffle); etc.
The 16-bit software was superseded by SuperFormula. Super Formula also calculates the Binomial Distribution Formula (BDF), the Binomial Standard Deviation (BSD), Statistical Standard Deviation (BSD) — and then some.

• Download Software: Science, Mathematics, Statistics, Lexicographic, Combinatorial: SuperFormula, FORMULA, OccupancySaliuParadox from the software downloads site - membership is necessary (a most reasonable fee).

I assembled all my mathematics, probability, statistics, combinatorics programs in a special software package named Scientia.

Read Ion Saliu's first book in print: Probability Theory, Live!
~ Discover profound philosophical implications of the Formula of TheEverything, including Mathematics, formulas, gambling, lottery, software, computer programming, logarithm function, the absurdity of God concept.

Resources in Theory of Probability, Mathematics, Statistics, Combinatorics, Software

See a comprehensive directory of the pages and materials on the subject of theory of probability, mathematics, statistics, combinatorics, plus software.

• Theory of Probability: Best introduction, formulae, algorithms, software.
• Bayes' Theorem, Conditional Probabilities, Simulation; Relation to Ion Saliu's Paradox.
• Birthday Paradox is a particular case of exponential sets (sets with duplicate elements); software to calculate, generate any form of Birthday Paradox cases.
• Probability: Pairing, Couple Swapping, Hypergeometric Distribution.
• Mathematics of Monty Hall Paradox, Classical Occupancy Problem; Ion Saliu's Paradox.
• Probability Player Index N Draws Ticket Number N.
• Calculate, Generate Exponents, Permutations (Factorial), Arrangements, Combinations.
• PI Day, Divine Proportion, Golden Proportion, Golden Number, PHI, Fibonacci.
• Censorship Ideology Religion Money Jealousy Piracy Hatred Ban Internet.
• Read all about the famous Fundamental Formula of Gambling.
• Almighty Number, Randomness, Universe, God, Order, History, Repetition.
  The Fundamental Formula of Gambling (FFG) may well be the Ultimate Formula of The Everything.
• Press Release: Mathematical Proof of the Absurdity of the God Concept.

The following pages at this website offer more special mathematical solutions, functions and formulas, especially in combinatorics. There are algorithms and special software to calculate and generate permutations, exponents, combinations, both for numbers and words. Also, lexicographical order or index can be easily calculated for a large variety of sets.

• Lexicographic, Lexicographical Order, Index, Ranking, Sets: Permutations, Exponential Sets, Combinations.
• Combination Lexicographic Order, Rank, Index: Comprehensive and Fast Algorithm.
• Combination Sequence Number or Lexicographic Order Debates.
• Combination Sequence Number or Lexicographic Order: Mathematical Algorithms.
• Combination1,2,3,4,5,6: Probability, Odds, Statistics.
• There are no naturally-occurring Geometric Shapes or Perfect Forms on Earth, indeed in the Universe, to justify a super intelligent, divine force in Cosmos.
  
• Only humans create perfect shapes (geometric as in geometry) in an attempt to have relative control over randomness.
• Download the best software.

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