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D’Alembert Law of Equilibruim Strategy

Among the various progressive betting systems used for casino table games today, more than a few were developed by some of the greatest scientific minds of the 18th century. Of special note, French mathematicians often took the lead in analyzing games of chance. They were fascinated with statistics and probability, particularly in connection with tossing coins and dice.

One of those fascinated French academics was Jean-Baptiste le Rond d’Alembert (1717~1783), whose credentials included honors in physics, philosophy and music theory as well as maths. One of his rather odd (an incorrect) theories was that the probability of a tossed coin landing “tails” would increase for every time that it came up “heads.” He referred to this as the “Law of Equilibrium.”

The mistake d’Alembert made was basing his theory solely upon observation rather than calculations. He had witnessed that whenever two events are equally likely, such as coin flips resulting in heads or tails, that they really do appear to occur in equal number over the long term. Streaks losses seem to be counterbalanced by streaks of wins.

In truth, of course, all coin flips are independent outcomes, unaffected by past results. Nevertheless, based upon his faulty logic, d’Alembert’s system of betting promoted a process of decreasing one’s bet when winning and increasing it when losing, confident that wins and losses would eventually become equal for wagers with odds of 1:1.

The d’Alembert Betting System is therefore most commonly used for “even money” bets, such as Red or Black wagers at the Roulette table, Pass or Don’t Pass wagers at the Craps table and Banker or Player wagers at the Baccarat table. With modification, the system can also be used for Blackjack as well as for sports betting where vigorous must be accounted for.

The objective of the d’Alembert Betting System is to win a single unit in profit, so the player begins by wagering one unit at Evens. Each time a bet succeeds, one unit will be subtracted from the total just wagered and the remainder will be the amount of the next bet. Whenever a bet loses, one unit is added to the total wagered for the next bet. The progression continues until the amount of the next wager becomes zero.

As an example, if £1 is the basic unit and the first bet succeeds, subtracting one unit results in a next wager of zero, so the progression ends with a profit of £1. On the other hand, if the bet loses, the wager is increased by one unit to £2. If it wins, reduce the wager by one unit back to £1. If it loses, increase it one unit to £3. Continue playing in this fashion until the required wager is zero, resulting in a single unit (£1) in profit. Then, the progression begins anew.

One aspect of the d’Alembert Betting System that sets it apart from Martingale is that it does not require risking huge amounts at unfavorable odds in an attempt to recover previously lost wagers. Also, it differs from Labouchere because strings of losses never increase the wager by more than a single unit for the next bet. The d’Alembert Betting System is thus a very slow and methodical approach to wagering, making it a less risky progression than its cousins.

Where the system fails, however, is in its basic premise of Equilibrium. An initial loss might easily be followed by series of wins and losses in equal number, never quite recovering the original wager. Indeed, the progression might never end at all.


The 8 most essential tips before visiting Las Vegas


  1. Visit mid-week. Hotel rooms often cost three to four times as much on the weekends as during the week!  Plus, with weekend crowds you’ll wait in line for everything.  Mid-week visitors save tons of money and have a much better time.
  2. Get a player’s card. Go to the Player’s Club desk at any casino you visit and sign up for a free player’s card, even if you don’t plan to gamble there.  Then the casino will mail you offers for discounted or even free rooms on your next visit.  You can also sign up for the MGM-Mirage card online, which will work at 10 Vegas Strip casinos.
  3. Set a strict bank limit and stick to it. Don’t bet more than you can afford to lose. Set a budget for each playing session and if you lose it, stop playing.
  4. Don’t play games you don’t understand.
  5. Tip the dealers $10/hr., and the cocktail waitresses $2 every drink or two.
  6. Use the buses. The Westcliff Express bus goes straight from the airport to the South Strip (MGM Grand, NY NY, Luxor, Tropicana) for $2.  The #108 goes from the airport to the Stratosphere and then to downtown, for $2.  Pay $7 for a 24-hour pass when you get on any bus, and then ride any bus in the city for free for the next 24 hours, including the Deuce and Strip/Downtown Express buses.
  7. Walk. The whole strip is four miles long and you could walk the whole thing in an hour and a half.  Plus, it helps work off the calories from the buffets.
  8. Bank your winnings. Set aside half your win when you win big, and never gamble it, so you have a guaranteed win.

Casino Probabilities : A must read intro



The casino roulette outcome was red 8 times in a row! What next?

Is black more likely on the next spin? No, it is not. Both red and black are equally likely. If you thought otherwise then the casinos love you, and you need to read this article right now.

In this article we’ll show here is why past events have no influence over future events. To understand this you need to know just a teeny tiny bit of math, and just one term, probability. Probability describes how likely it is that something will happen. There are three ways to refer to it: by fraction, by decimal, or by percentage. For example, say there are four cards, face-down, and you get to pick one. Three of them are aces. What are your chances of picking an ace? You have three chances out of four to get an ace. We can express this in any of these ways:

  • 3/4    (fraction)
  • 0.75   (decimal)
  • 75%   (percentage)

Each of these is just a different way of talking about the same thing. Notice that they’re pretty easy to convert, too. If you punch 3 divided by 4 into a calculator you get 0.75. And to convert a decimal to a percentage all you have to do is move the decimal two spaces to the right and add the percent sign. 0.75 is the same as 75%. What could be easier?

Okay, so now that we know how to refer to probabilities, let’s look at what they mean. Something that definitely will happen has a probability of 1 (or 1/1, or 100%, if you prefer). There’s a 100% chance the sun will come up tomorrow. Well, it’s not really a “chance” since it definitely will happen, but you get the idea. In our example of four cards, if all four were aces then your chances of picking an ace would be 4/4 = 1, it would definitely happen.

Something that definitely will not happen has a probability of 0. And in between 0 and 1 (or 0% to 100%) are all the things that could happen.

Your chances of winning some bet or series of bets might be 22%, 39%, 57%, or 83%. The higher the number, the more likely it will happen. Events over 50% will probably happen, events under 50% will probably not happen.

So far so good. So now let’s look at probability when an event happens many times, like flipping a coin over and over. The probability of getting heads on one flip is 1 out of 2 — one way to win out of two possible outcomes. We can call that 1/2 or 0.50 or 50%. But what are the chances of flipping the coin twice and getting heads both times? To figure this we multiply by the probability of each event:


First Flip
Second Flip

Of course, another way to express this is 50% x 50% = 25%.

Okay, so what are the chances of getting ten heads in a row?



Not very likely, of course.

So here’s where the gambler’s fallacy comes in: Say you’ve tossed the coin nine times and amazingly, you got nine heads. You figure that the next toss will be tails, because the probability of getting ten heads in a row is one in 1024, which is unlikely to happen!

The problem with this reasoning is that you’re not looking at the chances of getting ten heads in a row, you’re looking at the chances of getting one heads in a row. The heads that already happened no longer have a 50% chance of happening, they already happened, so their probability is 1. When you flip again the odds for that flip will be 50-50, same as it ever was.

Let’s introduce our hero, Mr. P, who will always be looking to the future to see what’s going to happen. He’s about to make ten coin flips, hoping to get ten heads. Here’s his outlook:



And here’s Mr. P. after flipping nine heads in a row, getting ready to make his tenth flip:



Now you’re saying, Hey, wait! How come all the 1/2’s turned into 1’s? The answer is that they’re no longer unknowns. Before you flip a coin you don’t know what’s going to happen so you have 50-50 odds. But after you flip the coin you definitely know what happened! After you flip a coin, the probability that you got a result is 1. You definitely flipped the coin. Definitely, definitely. So after you’ve flipped nine heads, the probability of flipping a tenth head is 1x1x1x1x1x1x1x1x1x 1/2 = 1/2.

Let’s have another look at Mr. P:



Notice that it doesn’t matter where on the table you stick him, the chances of his next flip being heads is always 1/2. Wherever he is, it doesn’t matter what happened before, his chances on his next toss are always 1 in 2.

How could it be otherwise? When you flip a coin you will get one result out of two possible outcomes. That’s 1 in 2, or 1/2. Why and how could those numbers change just because you got a bunch of heads or tails already? They couldn’t. The coin has no memory, it neither knows nor cares what was flipped before. If it’s a 1-out-of-2 coin, it will always be a 1-out-of-2 coin.

Still not convinced? Then here’s another way to think about it. Let’s say someone hands you a coin and asks, “What are the chances of flipping heads?” Without hesitation you’d probably say 1 out of 2? But wait a minute — if it were true that heads were more likely if tails has just come up a bunch of times, then why did you answer “1 in 2” right away when asked about the chances of getting heads? Why didn’t you say, “Well, you have to tell me whether tails has been coming up a lot before I can tell you whether heads has a fair shot or not.”? It’s simple: You didn’t ask about the previous flips because intuitively you know they’re unimportant. If someone hands you a coin, the chances of getting heads are 1 in 2, regardless of what happened before.

Would it really be the case that you answered “1 in 2,” and then your friend said, “Oh, I forgot to tell you, tails has just come up nine times in a row.” Would you now suddenly change your answer and say that heads is more likely? I hope not.

One last example: Let’s say your friend slides two quarters towards you across the table. He tells you that the first coin has been flipping normally, but the second quarter has just had nine tails in a row. Would you now believe that the chances of getting heads on the first coin are even but the chances of getting heads on the second coin are greater? Given two identical coins, could you really believe that one would be more likely to flip heads than the other? I hope not!

The same concept applies to roulette. An American roulette wheel has 18 red spots, 18 black spots, and 2 green spots. The chances of getting red on any one spin are 18/38. If you just saw nine reds in a row, what is the likelihood of getting black on the next spin?

18/38, same as it ever was.


Which casino game is more profitable?

his is one of the most often asked questions when it comes to online casinos. What game should I play to make the most money? Where am I likely to loose less.

The problem is that it is an almost impossible question to answer because casinos make it difficult to decide by changing the rules of the game while marketing them as the same.

Blackjack for example, has so many variants that it is hard to call blackjack the most profitable game of them all.

There is, Pontoon, Spanish 21, doubling down any number of cards, rescue, (or surrender), payout bonuses for five or more card 21’s, 6-7-8 21’s, 7-7-7 21’s, late surrender, and player blackjacks always winning and player 21. In Asia, the so called, Chinese Blackjack is very popular, (splitting is different).

Another popular game amongst online gamblers are slot machines, they can offer payouts ranging from 70% to 99%. Granted most well known online casinos would never offer a slot game that paid less than 95%. So that would make slots the most profitable game… if you knew in advance what the percentage payout was, many forums/websites claim to know the percentage, but one wonders how they arrived at that number in the first place, (the casinos will either lie or not give the actual payouts).

Because the payouts are not advertised it is very difficult for a user to reliably choose a slot.

Progressive slots also don’t pay as much because the casino has to build funds for the end bonus, (ranging from $10.000 to $1.000.000 and more).

Craps is also a mis-leading game, the “pass line” bet, which wins for a new shooter who rolls a 7 or 11, loses on a 2, 3, or 12, and on any other number requires him to roll that number (his point) again before rolling a 7, has an even money payoff that delivers a 1.41% edge to the house. The single-roll bets are just ridiculous: an ‘any 7’ bet pays 4:1 and gives the house a whopping 16% edge.

Roulette has two popular versions, the European version has 37 slots with a single 0; the American version has an extra slot, a 00 to make 38.

The house advantage is 2.7% For European and 5.26% for the American table.

So the choice is simple if you must play roulette, play European!

But as a whole roulette is not such a safe bet, (and some ‘sure way’ techniques like Martingale method make is downright dangerous to play).

They change the rules all the time.

Remember also that certain rules changes are employed to create new variant games.

These changes actually increase the house edge in these games. But they are cleverly worded to give the opposite impression to the unsuspecting players.

Double Exposure Blackjack is a variant in which the dealer’s cards are both face-up.

This game increases house edge by paying even money on blackjacks and players losing ties.

Double Attack Blackjack has very liberal blackjack rules and the option of increasing one’s wager after seeing the dealer’s up card.

And the winner is?

Classic blackjack in most its forms is usually the game that offers the best returns.

With correct basic strategy, a Spanish 21 almost always has a higher house edge than any comparable Blackjack game.


The secrets of the roulette

One or double zero

There are two types of roulette: With one zero and with double zero. The edge of the Casino is greater with double zero. The intensity of the casino’s edge varies, but single zero is the cheapest in which case you only loose half your betting on the chances red/black, odd/even and high/low when zero is the one. Sometimes you loose your entire betting – this is important to remember before starting.

Payouts on the different chances are:

1 nummer (plein) 35:1
2 numbers (cheval) 17:1
Transversal plein 3 numbers 11:1
Carre 4 numbers 8:1
Transversale simple 6 numbers 5:1
Dozen/columns 2:1
Single chances 1:1


1 number has a probability of 1:37 and since the pay out is 1:35, the casino has secured itself a nice, easy profit. As a point of departure, you cannot chance these playing normally – not even when you use progressions or gamble on extreme probabilities. For example to await that a red has come six times in a row and then betting on black. Should it not come the first time you double from 1 to 2 units. And you continue this way from 2 to 4 etc. This is known as a Martingale. It is a safe route to ruin, even though you only need one win to be back to +1 unit.

Chart of series

There are several arguments against this method. Why sacrifice 4, 8, 16, 32, 64 or maybe 512 units to win one unit? If you want to play with varying stake money, then why choose a technique where the stake money is more moderate, but requires several hits. Due to the 50/50 chance of red/black the serie spread looks as follows:

2 series on 10 or more than 10
2 series on 9
4 series on 8
8 series on 7
16 series on 6
32 series on 5
64 series on 4
128 series on 3
256 series on 2
512 series on 1

You can start anywhere in the chart and see that there are just as many series on e.g. 6 than there are series longer than 6. You might as well bet on a series continuing rather than ending. The probability is the same. Many choose to continue with a series, flat bet, as long as it lasts arguing that you cannot know who long it is, but that it would be frustrating to get off early in a long series.

Others choose a different more intuitive strategy. You select a player, who is in a stream of bad luck (perhaps after having won for a while) and bet the exact opposite. There are people, who just aren’t very lucky. If you can find one of those and play the opposite…well…sometimes it works.

Some tips

A couple of advice: never bet too much in one spin so that it matters whether you win or loose.

Never bring your credit card. Decide on a fixed limit and bring it in cash – leave, if you have lost it all.

Should you win – it is a possibility – a good rule is only to give back part of your winnings to the casino so to speak. Let’s say you won 300…then stash away 100 or 200 or cash it and play with the remainder. Are you in a winning stream then cash it as soon as you can and continue with the remaining 1/3. This leaves you a little “buffer” should your winning stream stagnate.


  • Set a fixed limit that you can afford. Do not change, bend or alter this rule!
  • Never bet so much in one spin that it matters if you win or loose.
  • Continue to stash or cash parts of the winnings and use the remainder to pursue your winning stream.

You basically cannot beat the casino by numbers alone. Casinos always pay back less than they get in. It’s what they live of.

Deterministic vs. chaotic

The only element likely to change the probabilities is observation of physical events of a Newtonian nature with side effects of deterministic and chaotic nature. In this case, the effect of chaotic nature is less than the effect of the deterministic. That means if you transform the powers in the ballistic process of the ball and the speed of the rotor, you can predict the point, where gravity defeats the centrifugal power, and the ball begins to fall. In this case the deterministic part of the spin defeats the chaotic, where the ball hits one or more diamonds and begins to jump until stopping. After a long period of practicing, you can visualize the deterministic part and with statistics you can overcome the chaotic part, which often is purely chaotic. This is called Visual Tracking. Measured in one spin: the chaotic part is chaotic and all 37 numbers have a chance, but the chaotic part can be analyzed and the physics behind is far from as chaotic as it appears, and the jump length of the ball is not random. If it is (and there are wheels where that is the case) it is a wheel that is completely random.


Another option is a wheel with a defect – or bias – making numbers or sectors of the wheel more probable. Casinos keep a close watch for this scenario and the chance of you finding it by yourself before they do is unlikely today. Some years back teams went from casino to casino noting down all the number on all wheels. For example Dr. Richard Jarecki who on a regular basis robbed San Remo casino leading to their temporary closure. An old story unlikely to repeat itself today due to tight surveillance. Another famous one is Benno Winkel who with a team of paid writers went around and played wheels with provable biases. That’s biases that had more than 3 standard deviations. He lost everything later on what has lead to many believing that his game was not based on true bias, but rather betting favorites, which later caught up with him due to elementary probability theory.