My Latest Conundrum

MY last post got me thinking about the difference between statistics and probability or expected outcomes vs. actual outcomes.

According to the law of big numbers, a large sample of random events will tend to move toward expected outcomes. Therefore, if you were to look at 38 million roulette decisions, you should see that each number appeared about 1 million times and that about 47% of the decisions would be red for example.

Now, IF this proposition is true then it is fair to say that the average gap between repeats would have to be about 38 spins. This means that if you were to take any particular number, like the number 3 for example and see how often it appears, the answer would be once every 38 spins.

IF it is true that numbers tend to appear every 38 spins in the long run, it would seem to me that we have an exploitable event.

Suppose you place one chip valued at $18 on the "high" numbers (outside bet on numbers 19-36), probability and statistics would show that you would probably win this bet almost half the time (because you have almost half the layout covered). Now suppose instead, you placed 18 individual bets on the inside placing one dollar chips on each of the numbers between 19 and 36. Statistics and probability would tell you that you should win almost half of these bets and in fact the results should be identical to the "other" way mentioned above.

Now lets look at a moving target. Suppose you were to place 18 individual one-dollar bets on the 18 numbers which are NOT the last 18 numbers to appear (not considering the zeros just for the sake of this example). Let me re-phrase that. Make a list of the last 18 numbers to appear and place 18 one-dollar bets on the "other" numbers. Would this provide you an advantage? Should you win this bet MORE OFTEN than half of the time?

Now looking back at our first proposition. IF IT IS TRUE that numbers tend to appear every 38 spins then it would seem that while many numbers appear less often than 18 spins many more numbers are appearing at intervals greater than 18 spins. (If the average gap between repeats is 38, then necessarily some number gaps would be greater than 38.)

NOW, IF the average gap is greater than 18, then wouldn't it make sense that by playing 18 numbers which are not the last 18, would give you an advantage? Wouldn't it seem that the next number to come up is more likely to NOT be of the last 18 than TO be of the last 18.

And if so, doesn't this fly in the face of the fact in my most recent thread that the last 38 decisions is likely to only be comprised of 24 numbers??

Something to think about . . .

PRELIMINARY TESTING

I took a look at some spins created by a random number generator. The sample is entirely too small to draw any conclusions, but the results are encouraging.

The First sample i looked at was 1000 spins. I recorded data for 982 spins (skipping the first 18). I looked at each spin and measured the distance since the last appearance of that number. If the number was 18 or less, I counted this as a loss of 20 units. If the number was greater than 18, I counted that as a win of 16 units. The explanation is as follows: if you bet 1 unit on each of the 20 numbers that do not represent the last 18 numbers to show, then you will win a net of 16 units if the next number is not one of the last 18 and you will lose 20 units if the number is among the last 18.

This produced +1960 units for the 982 spin sample. The gain is 2 units per spin (or about than 1/10th of the minimum bet).

If you play with $1 chips, you would need to bet $20 a spin. Each win will net you $16. You would have won $2 per spin or $120 per hour.

If you play with $5 chips, you would need to bet $100 a spin. You would have won $600 per hour.

Not bad at all.

What about drawdowns? I broke my preliminary batch of numbers into hour long sessions. Only 2 sessions ended up in the negative and they were -72 units. I would feel comfortable with a bankroll of 300 units or 400 units.

SECOND BATCH

The second batch of randomly generated numbers represented 282 spins. The total gain was 660 units (2.34 per spin) or slightly higher than the first batch average (2 units per spin). Perhaps the best way to look at this is a gain of about 1/10th unit per spin or 6 units per 60 spins or 6u/hour.

(I remind anyone who reads this that the preliminary test samples are very small at this point.)

THIRD BATCH

The 3rd batch was 982 spins which produced 1622 units or 1.65 units per spin.

Totals through 3 batches:

2,246 spins = 4,242 units or 1.89 units per spin (113.4 units per hour).

FOURTH BATCH

982 spins with a total gain of 2212 or 2.25 units per spin.

Totals through 4 batches:

3,228 spins = 6,454 units or 1.999 units per spin

Average = 120 units per hour (based on 60 spins per hour)

FIFTH BATCH

982 spins with a total gain of 2052 or 2.09 units per spin.

Totals through 5 batches:

4,210 spins = 8,506 units or 2.02 units per spin

Average = 121.2 units per hour

SIXTH BATCH (abnormally high results)

456 spins with a total gain of 1860 units or 4.078 units per spin.

Totals through 6 batches:

4,666 spins = 10,366 units or 2.22 units per spin

Average = 133.2 units per hour.

SEVENTH BATCH

979 spins with a total gain of 1947 units or 1.988 units per spin.

Totals through 7 batches:

5,645 spins = 12,313 units or 2.18 units per spin

(very impressive through 5,645 spins*)

*but see "THE ERROR" below

PRACTICAL CONSIDERATIONS

I have to admit that covering 20 numbers within a few seconds may be challenging. The task is not only to cover 20 numbers but to select those numbers as well. The key is to cover all except the last 18 to show. With a partner, one could be placing all the numbers as the other person reads the board and removes the most recent 18. If this works, I'm sure I can find a way to get the coverage I need.

Another idea might be to have a small laminated copy of the layout and to "x" out the last eighteen numbers with a dry-erase pen. Then use this to show you where to place your bets. Another thought would be that you do not need to place very bet, it would make sense that your chances of winning are the same if you placed a bet every other spin. So, if it took you a while to get your info together, you could just play a table minimum outside bet on one of the even chances and then place your 20 inside bets on the next spin.

The above numbers are actually quite staggering. Using $1 chips, you could expect to win 130 units per hour. Using $5 chips, you could expect to win $654 per hour. All with a bankroll of only 400 chips ($400 or $2,000).

The worst-case scenario (in the first 7 batches tested) is an hour with losses of 288 units ($288 or $1,440 using $5 chips).

The best hour (in the first 7 batches tested) is +396 units ($396 or $1,980).

ONE POTENTIAL FLAW

One potential flaw in my test numbers is that I am counting as wins any number that has not appeared in the last 18 spins. I am covering 20 numbers and leaving the last 18 uncovered. However, if the last 18 spins include any repeats, then 20 numbers would not be enough coverage. For example, 18 spins might only represent 14 numbers, if so, then I would have to cover 24 numbers to cover all numbers not included in the last 18 spins. This would mean that the only ideal time to bet is when the last 18 spins have no repeats.

THE ERROR (OF MY WAYS)

I've come to accept that the "one potential flaw" mentioned above is in fact the critical flaw of my testing. The very impressive numbers above are based on a scoring system where I assigned a +16 to each win (repeat gap larger than 18) and a -20 to each loss (repeat gap 18 or less). The problem is that this would only be accurate if the last 18 spins represented exactly 18 distinct numbers (which should rarely be the case). In order to truly test this tehory, I would have to look at the last 18 numbers to appear (not the last 18 spins). The question becomse, how many spins does it take to produce 18 numbers, my guess is something like 21 or 22. If I were to go back and re-score the test numbers using 21 or 22 spins, the result would be dramatically different (obviously worse). A different way to look at the test nukbers is that 18 spins probably only represents about 16 numbers on average (maybe even less). So, to cover all the other numbers, I would have to place 22 bets. This means each win would be only a +14 and each loss would be a -22. No doubt this would also drastically reduce the results above.

SO

So, does the law of big numbers help us to find an exploitable anomoly? Is the average gap between hits really 38 or is it more like around 24?

Back to the drawing board . . .

Systems 101 - A Primer

It dawned on me while reading a message board that there are a lot of beginner-level players out there searching the web for some useful information and often unable to follow the threads because the more experienced players seem to be writing in code.

I thought I'd begin a thread design for the entry-level gambler looking for a system to play. You will not find any systems in this thread only basic information. You will also not find the rules of play here. If you will check other threads within this blog, I have recommended some books for the beginner (and intermediate) player.

BS v MM

I think a good place to start is the distinction between bet selection and money management. Simply put, Bet Selection systems (BS) tell the player where to place the next bet (like on red, or player, or passline for example). Money Management (MM) systems tell the player how much to bet on the next decision.

Most systems are mathematical templates that tell the player when to bet, where to bet and how much to bet. Typically systems are designed for Even Chance or 50/50 decisions (EC).

The House Edge

It should come as no surprise that ALL Casino games have a built in house edge. This is the amount of money one can expect to lose by playing the game. The edge is determined by the difference between the true odds and the payout. A simple example is playing one number straight up in roulette. If you place one unit on the 17 and the 17 hits, you are paid 35 to 1. You receive 36 units (comprised of your original bet and 35 of the house's chips, now might be a good time to stop playing). Because there are 38 numbers on an American roulette wheel, you have a one in 38 chance of winning a bet that pays 35 to 1. If true odds were paid, you would be paid 37 house units instead of 35. The house edge changes from game to game, but this simple illustration would be true for every single bet you can place in the casino. Some people win, some people lose but the casino ALWAYS wins, just look at the casino and this fact should be plainly evident.

MATH and TESTING

It is widely claimed and typically accepted that math proves that systems do not work. It seems that with thorough testing, all that one can hope for is to break even (or more accurately, to break even less the house edge).

There are system testers available for system's players to test their theories against actual casino results. (see another thread in this blog for links to purchase testers). The idea behind testing is that if your system can beat the testers, then it should perform strongly under real circumstances. It should seem obvious that if you can not beat the testers, you system has a good chance of failing in the casino.

Is there hope? If math brings you to the conclusion that systems will not win and the testers are nearly impossible to beat, is there hope for developing a successful strategy? Many successful players claim that some systems perform reliably in the short run and the key to winning in the long run is changing systems to respond to the game as you play it. Testing several systems which are triggered by (sometimes) subtle changes in the game is a very difficult task. Therefore, it is plausible that a successful systems player could win in the long term by making changes that would not be easily duplicated in the tester books (like leaving the table in search of a more lively one for example).

LONG RUN v. SHORT RUN

It is important to understand that series of random events tend to perform in accordance with their expected mathematical probabilities in the long run BUT rarely do in the short run. If an event has a near 50% likelihood of occurring (like the "player" winning a hand at baccarat for example), then if you looked at a large sample (like 1,000 decisions) you would probably find the even occurs very close to 50% of the time. This can be relied upon in the long run. However, it might be unwise to bet in anticipation of a 50% occurrence in the short run (like the next 6 decisions for example).

There is another concept to throw into the mix. That is the "standard deviation", but for this "beginner level" primer, I do not think it is necessary to go into how it works, Just be aware that when looking at a set of decisions, they can be expected to perform "close to" their expectation and there is a mathematical way to determine how "close to" the expectation would be normal (or at what point the numbers would be abnormal) and this is called the standard deviation.

OTHER ABBREVIATIONS

FTL = Follow The Last. This is a bet selection system that simply means your next bet is that the last decision will repeat. If red hits on roulette, your next bet is on red.

OLD = Opposite Last Decision. This of course is the opposite of FTL, if red hits on Roulette, your next bet should be black.

DBL = Decision Before Last. Here you would bet the same as the decision before the last decision. This simple Bet Selection system has the benefit of breaking up streaks that could work against you. (I'll try to come back and present an example of this here later.)

more later . . .

The "Cycle"

THE CYCLE

Part I

Lately I've been focusing my work on a concept I call the "Cycle."

I have posted elsewhere my idea of "going for half" and these two concepts work well together.

I have also written about mathematical expectancy and this is a good place to begin an explanation of the cycle.

All gambling propositions have a probability which can be described in the form of mathematical expectancy. A very simple example would be betting one number, 17 for example, straight up on an American roulette wheel. Since there are 38 numbers on an American roulette wheel, it is said that the probability of the number 17 hitting on the next spin is 1 in 38. The mathematical expectation is that we can "expect" a hit on the number 17 once in 38 spins. The "cycle" for this proposition therefore is 38 spins.

The simple example above is provided merely to illustrate the terminology. The concept becomes a little more complicated when we look at more complex bets, like betting 2 dozens and 2 columns for example, or using progressions.

We all know that the so-called even-money outside bets (like red/black for example) are close to 50/50 propositions. We also know that when you factor in the house edge, your chances of winning any particular "even-money" bet is a little less than 50%. In short, the "odds" are against you or in other words, you are more likely to lose this bet than to win this bet.

We also know that you can place bets that you are more likely to win but that the payoff is less than one to one. For example: Playing 2 columns gives you 24 of 38 chances to win, however the payoff is 1 to 2, you will be wagering two units in hopes of netting one.

My theory about "maximum coverage bets" (and I hope to come up with a better term than that) is that when you employ a progression, your chances of losing your series is drastically reduced.

NONE of this defeats the house edge I remind you. But, I accept that cold fact with all systems.

What I hope to develop here is a way of looking at cycles and maximum coverage bets to allow us to chose bets that will produce small but steady gains with rare losses (which will necessarily be large).

More later . . . .

The Cycle Within a Cycle

Using multiple levels of progression, leads to bigger cycles containing smaller cycles. For example: Suppose your bet was a three step martingale. You are betting on Red and you bet one unit on your first bet, then double after a loss, and again. Your progression is 1 2 4. Each winning series results in a gain of 1 unit. Each losing series results in a loss of 7 units. We know that you can expect to lose a series about once in 8 series. Assuming for this discussion that you are playing a true 50/50 game, a wheel with no house numbers, a wheel with exactly half of the numbers being red. Under these circumstances, you can expect to win 7 series and lose one. This is the first level of progression.

Now suppose you decide to add another level of martingale. After a losing series, your first bet will be 2 and your progression will be 2 4 8. After a win, you will return to your original series.

Now look at the cycle. Y0u have a cycle of 8 series where you can expect to win 7 series and lose one to break even for the cycle. This cycle can be expected to take 24 spins or decisions. By adding the 2nd level martingale, you are increasing your net by +1. IF you experience the mathematical expectancy of a typical cycle, you will end your 24 spins up one unit instead of break even.

OF COURSE, there is another mathematical expectancy of losing back to back series. This other expectancy has another point in a larger cycle where you can expect to be brought back to zero or even (or to a negative amount equal to the house edge) . In the original progression we saw that we can expect to lose one series out of eight. We then added a second level of progression gambling that we would not encounter our one in eight losses back-to-back. How often will that happen? [I have notes elsewhere and I'll return to fill in this gap] This would be the bigger cycle. Eventually, you can expect to be brought back to even (or zero) when the bigger cycle runs its course. By adding yet greater levels of progression you are increasing the size of the cycle and it is my theory that you are increasing the amount of time you can expect to be ahead of the game before being brought back to zero. AND MAYBE - if you have several tactics for stretching out the cycle of expectancy, then you can quit while ahead more often OR change strategies while ahead in the cycle.

More later . . .

The "No Lose" Expectancy

(Which, of course WILL Lose as expected)

As a general illustration of the discussion so far, I offer this example:

For this example, we are playing a wheel which produces 60 decisions an hour. We know that we can develop a system that has an expected loss of one time in 60 decisions. This one loss would be expected to eliminate all winnings from the cycle of 59 wins. If we play this game for only 30 minutes and IF we are ahead at 30 minutes, then we can quit under my notion of "going of half." The question then becomes, of all the 30 minutes sessions that we will play, how many will include the dreaded losing decision. Math would probably tell us half. Real play may show us something different. We know that IF we have one more winning 30 minute session than losing 30 minute session, then we'd be ahead in the big cycle. And if the sessions were kind enough to come evenly spread out, you'd always be only 30 minutes away from being ahead.

If we strip this theory down to it's simplest form, it becomes WAY less attractive. Yet there is something about the more complicated version that I find appealing.

Here is the stripped down version: Suppose we are playing a true 50/50 game and the game produced its mathematically expected results with regularity in the short run. So that if you flat-bet and you encountered a win/loss series like this: W L W L W L W L W L, you would always be just one or two decisions away from a profit. Following through with our example above, you could always quit while ahead and it would be easy to do so.

We find this to be unappealing because we know that Roulette and Baccarat and other near 50/50 games do not produce reliable results in the short run. We know that the 50/50 game is very volatile and that it takes THOUSANDS of spins or decisions for the results to approach the expected 50/50 mark.

My theory is that the smaller the cycle, the more volatile and unpredictable the game. BUT on the other hand, the larger the cycle, the more predictable the game becomes.

I found an Excellent article and example of a No Lose Expectancy System, I'll post a link here when i get my hands on it again.

More Later . . .