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The Reading-room [Info]A Discussion of Certain Common StrategiesAuthor: Bryan HoldenThe conclusions I am going to draw from this discussion are, to say the least, somewhat counter-intuitive. Therefore I ask that you suspend your disbelief for a while and follow the logic through. It may be that my logic is flawed in some way, in which case I would appreciate any feedback, but I have been through it quite closely and it seems conclusive to me. The situations under consideration are all those where only two outcomes are involved. Such as : coin tossing, baccarat, even-money roulette, and so on. As this is a theoretical discussion only, in the case of even-money roulette I am ignoring the effect of the zero(es) and in baccarat the slight advantage of Banker over Player. For purposes of this discussion I will use coin tossing, and refer to heads as "H" and tails as "T", but obviously I could just as easily use "B" and "P" from baccarat or "R" and "B" from roulette. I will start with a discussion of "streaks". What do I mean by this? Well, we all know that when tossing a coin a bunch of times we don't get an infinite series of "HTHTHTH...", but rather a discontinuous series of H's and T's in a seemingly random fashion, hence we must, at some times, get back-to-back H's and back- to-back T's as in, for example, the following series - HTHHTTTTHTHHH. In this example we have the following (in order) - a streak of H's of length one (yes, a streak CAN be of length one - by definition), a streak of T's length one, H's length two, T's length four, H's length one, T's length one, and H's length three. Note that we cannot really say that the last streak is length three. Since a T has not yet terminated it, all we can really say is that it is a streak of AT LEAST length three. This is an important principle - its start and termination points determine a streak. This might seem a little obvious but it is nonetheless important. This is because the length of a streak can only be determined when it ends, and it ends only when the opposite outcome arrives. A corollary to this is, therefore, that the end point (or terminator) for a streak is also the starter for the next streak. Or, to put it another way (which is more relevant), the starter for a streak is also the terminator for the previous streak. This is important - we will return to this principle later. From the analysis of the above example, and excluding the last streak which is unterminated, we can produce the following table. The first column is the length of the streak and the second the number of times a streak of that length occurred, for a total of 6 streaks in all. 1 4 2 1 3 0 4 1 (Table I)This is a conventional method for recording streaks and can be found in other common publications such as "72 Days at the Baccarat Table" by Erick St. Germain. However, for purposes of this discussion we are not concerned about whether the streak was one comprising all H's or all T's. The first major pointNow that I have established our frame of reference, let's get into it!Let us say that a T has been tossed. We don't as yet know how long this streak is going to be (remember? no termination point yet). For it to be length one, the next toss must be an H. This would terminate the streak and determine it to be length one. If, however, the next toss produces another T then we have not yet terminated the streak and we are in the position of being able to say only that the streak is AT LEAST length two. So let's go back to that first toss after the initial T. We know that there is a 50-50 chance of an H so, over a statistically significant sample, 50% of all the streaks will be length one. Now this is the first major point in this discussion so it is important that you think about this and, hopefully, agree with me. Let's recap - we have an initial toss of a T; the next toss will either be a T or an H; there is a 50% chance of it being either one; therefore half of the streaks in our statistically significant sample must be of length one since 50% of the tosses will produce a T. OK? I hope so, because if you don't agree with this then we are done! If you don't agree then perhaps you could explain to me why - I would be happy to hear any counter-arguments. Before you argue about deviations from the norm and all that, let me say that I am aware of the fact that no sample will produce EXACTLY this result. Real-life just isn't like that is it? However, I do contend that the result of any statistically significant sample will definitely TEND TOWARDS this result. Extension of the first major pointSo, assuming you are with me so far, we come to the next stage in the discussion.br> Of all the streaks in our sample, we have found that 50% of them will be of length one. Hence this leaves 50% still undetermined, but these must be AT LEAST length two. Therefore, in our example, we have "TT" so far - that is, a streak of T's is in progress with length yet to be determined. Now all we do is extend the previous argument and state that "50% of the remainder will terminate and 50% will not". Hence we can say that 50% of the remaining streaks will be length 2. So, for example, if we had a sample that contained 64 streaks in total, we can say that 32 of them will be length one, leaving 32 to be greater than length one, and 16 of these remaining 32 will be length two. This extension continues on the remainder of the remainder and so on, of course, so that we end up with a theoretical table such as follows - 1 32 2 16 3 8 4 4 5 2 6 1 (Table II)Note that this actually adds up to 65 streaks in total - this is just a result of "binary chopping". We could have stopped at length 5 but this would not be strictly correct since 2 CAN be divided by 2, so let us just say that we started out with 65 and not worry too much about the uneven division by 2. Real-life doesn't care anyway since you can't have a streak of length 2.5 for example, and furthermore real-life won't be bound by statistically perfect expectations - we are looking for what real-life is going to tend towards. ExampleThe following table has been extracted from the aforementioned book "72 Days...".For purposes of this example I have applied the principle stated in the Frame of Reference chapter, that is, a streak has to be terminated before its' length can be determined. Hence I have treated the data as a continuous stream (ignoring Ties of course). For example, the last streak in Shoe # 1 is indeed length three since the first outcome of Shoe # 2 is a Player which effectively terminates the streak of Bankers at the end of Shoe # 1. On the other hand, Shoe # 2 ends with a single Banker and Shoe # 3 starts with a run of 2 Bankers and so this is counted as a single streak of Bankers of length three. Furthermore, I have ignored the last streak of Shoe # 600 (one Banker) since it is unknown what length it would have been if it had been properly terminated. By the above definitions, there are 20594 streaks in total. The actual result from the data in the book is in column 2 and the theoretical breakdown of this number of streaks is in column 3 for comparison. 1 10344 10297 2 5142 5148 3 2549 2574 4 1305 1287 5 623 644 6 310 322 7 174 161 8 81 81 9 29 40 10 18 20 11 10 10 12 4 5 13 1 3 14 3 1 15 1 1 Tot. 20594 20594 (Table III)As you can see, the actual result is remarkably close to the statistical expectation. (Aside - a common myth is that Baccarat is a "streaky" game. This is nonsense, as you can see from the above table.) The crux of the matterNow that we have arrived at the same playing field, let's see where this takes us in terms of the original purpose of this discussion (yes, we are finally getting there!).We have a little bit of thinking to do now because this next stage takes a somewhat lateral approach. What I want to do is compare a number of strategies using the same approach. By "strategies" I mean predictions (or bets if you are a gambler) for the next outcome. These strategies are : same as last decision (LD), opposite of last decision (OL), decision before last (DBL), and opposite of decision before last (ODBL). In other words, what would be the final total result if you always bet that the next outcome would be : "the same as the last outcome", or "the opposite of the last outcome", or "the same as the outcome before last", or "the opposite of the outcome before last". Since we are analysing using the "streaks" approach we need to consider the result of each of these strategies for each streak length. 1. Same as last decision (LD).
To generalise for this particular case (same as last) we always have I fail and a number of successes equal to the length of the streak minus 1. 2. Opposite of last decision (OL).
Hence, we can extrapolate the other results as : 1 success and a number of fails equal to the length of the streak minus 1. 3. Decision before last (DBL).This one is a bit more difficult to visualise. You have to cast your mind back one more outcome to determine what the prediction is going to be. If we use the same starter as above (T) then the outcome before that must have been an H. Remember that principle stated earlier about starters and terminators? "A starter is also the terminator for the previous streak." Therefore, our starter T must have been the terminator for the previous streak and so the previous streak must have been H's. Hence the outcome before our T must have been an H. The following analysis assumes this.
We can now extrapolate and see that for streaks of length two or greater we have 2 fails and a number of successes equal to the length of the streak minus 2. 4. Opposite of decision before last (ODBL).This turns out to be the opposite of the above (pretty much as we would expect really).
The extrapolation is, therefore, for streaks of length two or greater we have 2 successes and a number of fails equal to the length of the streak minus 2. With me so far? Now we can start to wrap this all up and draw some conclusions. The final tablesWe are now in a position to draw up a table of comparisons using the data and principles developed above.The tables that follow are an extension of the simple ones that I have included previously. That is, the length of the streak is the left-most column, and column 2 is the number of streaks of length as stated in column 1. The third column is the number of outcomes involved in the streaks and so is actually column 2 multiplied by column 1. The remaining columns are in groups of 2, each pair representing one of the strategies discussed above (such as "last decision" and "decision before last"). Each pair is further broken down into "successes" and "fails", as derived above. Totals are displayed at the bottom. Table IV is an analysis of data from the theoretical result (column 3) of Table III. Stk Nbr Nbr - LD - - OL - - DBL - - ODBL - Lgth Stks O/cm's Succ Fail Succ Fail Succ Fail Succ Fail -------------------------------------------------------------------------- 1 10297 10297 0 10297 10297 0 10297 0 0 10297 2 5148 10296 5148 5148 5148 5148 0 10296 10296 0 3 2574 7722 5148 2574 2574 5148 2574 5148 5148 2574 4 1287 5148 3861 1287 1287 3861 2574 2574 2574 2574 5 644 3220 2576 644 644 2576 1932 1288 1288 1932 6 322 1932 1610 322 322 1610 1288 644 644 1288 7 161 1127 966 161 161 966 805 322 322 805 8 81 648 567 81 81 567 486 162 162 486 9 40 360 320 40 40 320 280 80 80 280 10 20 200 180 20 20 180 160 40 40 160 11 10 110 100 10 10 100 90 20 20 90 12 5 60 55 5 5 55 50 10 10 50 13 3 39 36 3 3 36 33 6 6 33 14 1 14 13 1 1 13 12 2 2 12 15 1 15 14 1 1 14 13 2 2 13 Tot. 20594 41188 20594 20594 20594 20594 20594 20594 20594 20594 (Table IV) Table V is an analysis of data from the actual result (column 2) of Table III. Stk Nbr Nbr - LD - - OL - - DBL - - ODBL - Lgth Stks O/cm's Succ Fail Succ Fail Succ Fail Succ Fail -------------------------------------------------------------------------- 1 10344 10344 0 10344 10344 0 10344 0 0 10344 2 5142 10284 5142 5142 5142 5142 0 10284 10284 0 3 2549 7647 5098 2549 2549 5098 2549 5098 5098 2549 4 1305 5220 3915 1305 1305 3915 2610 2610 2610 2610 5 623 3115 2492 623 623 2492 1869 1246 1246 1869 6 310 1860 1550 310 310 1550 1240 620 620 1240 7 174 1218 1044 174 174 1044 870 348 348 870 8 81 648 567 81 81 567 486 162 162 486 9 29 261 232 29 29 232 203 58 58 203 10 18 180 162 18 18 162 144 36 36 144 11 10 110 100 10 10 100 90 20 20 90 12 4 48 44 4 4 44 40 8 8 40 13 1 13 12 1 1 12 11 2 2 11 14 3 42 39 3 3 39 36 6 6 36 15 1 15 14 1 1 14 13 2 2 13 Tot. 20594 41005 20411 20594 20594 20411 20505 20500 20500 20505 (Table V) DiscussionCan we draw any conclusions from all this?Well, yes, I think we can. Looking at Table IV (the theoretical result) the most obvious one is that following any of these strategies gives no real advantage over time. It doesn't matter whether you follow the last decision or the decision before last (the two most popular ones), there is no advantage gained over the long term. This is, of course, devastating news to all those system followers who advocate the use of one or the other - and there are a lot of them. After all, it SEEMS sensible to "follow the trend" which is really just the last decision, or, if you want to be a bit more sophisticated, follow the decision before last which captures a proportion of two types of trend (all the same, or alternating). But what about the ACTUAL data from "72 Days..." (Table V). If you look at that table carefully you will see that OL appears to give an advantage of (20594 - 20411) / 41005 x 100 = 0.446%, and DBL gives a very slight (smaller than OL) advantage also. What does this mean? Well, we know that real-life deviates from the norm on occasions, but we also know that over time it SWINGS ABOUT the norm. Here we have a perfect example of this deviation from the norm and one conclusion that can be drawn from this is that even in a sample as large as 600 shoes (over 40,000 relevant decisions) this deviation can show up in one direction or the other - in fact I would hazard to say, it MUST. What would truly surprise me would be the results coming out EXACTLY ON the norm. So if it is unlikely for this "on the norm" to occur, then the actual result MUST be on one side or the other of it. Therefore, another sample of the same size could just as readily show the opposite trend (ie. be on the other side of the norm). So, the apparent advantage shown by this actual data table is an illusion. There is no such advantage shown in the theoretical table. Hence, any system that tries to capitalize on a small advantage such as that shown in this particular data set, will necessarily be disadvantaged when the swing occurs and the trend shifts over to the other side of the norm. Furthermore, since every data set is guaranteed to show deviances such as this (or, remotely possibly, be exactly ON the norm in which case there is no advantage shown), there is no "sufficiently large" data sample that can ever validate such a strategy. ConclusionsThe conclusions that I have drawn from all of the above are -
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