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| The Odds Calculators: Partial simulations vs. compact formulas |
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So-called odds calculators have spread all over the Internet, whether as interactive software programs or just as odds charts and tables. These programs have dedicated sections on most gambling websites, on the pretense that they offer the final winning odds or the odds of reaching a certain game configuration at a certain moment of the game. In reality, these sites do not deliver what they promise, namely odds and probabilities, as we shall see further. Most of these odds calculators are based on partial computer simulations. For convenience, we will abbreviate them as OCBPSs (odds calculators based on partial simulations). How a partial simulation works Each program reproduces on a computer the game under discussion (usually, a casino game) by using in its code all parameters that describe its rules, initial conditions, and progress. Both external and internal users can simulate the game through one of these programs and some of the data from every simulation are recorded. Specific gaming situations and certain events related to these situations are then tracked. A certain gaming situation that requires the display of some statistical data is tracked and recorded each time it is encountered during the simulations. For the respective situation, the occurrence of the proposed event is also recorded in binary mode (1 if the event occurred, and 0 if not). The OCBPS database is updated with every simulation, and what the program returns after each simulation is the ratio between the number of occurrences of the tracked event and the entire number of simulations that match the respective gaming situation until that moment. What is a partial simulation from probability point of view Now let us talk about where mathematics frames the partial simulation as an experiment generating probability-measurable events. If we refer to probability as limit of a sequence of relative frequencies, as the Law of Large Numbers states, then an OCBPS builds the sequence of independent experiments having outcomes statistically recorded through its simulations for the respective game. The experiments that match the given gaming situation form a subsequence of that first sequence of experiments. Therefore, what an OCBPS returns is not the probability of the predicted event, but merely the terms of the sequence of relative frequencies of its occurrences, in increasing rank. The inputs and the returns of an OCBPS are what statisticians call a practical statistic, which are also a partial statistic, and not probabilities (odds) as they try to lead users to believe. The difference is similar to the one between a sequence's term and sequence's limit. This difference can be at any value. Of course, as the rank increases, the difference diminishes. What, then, are the contradictions in an OCBPS as concerns its internal structure and its usage? â€" The main contradiction is a theoretical one: the program is not an odds calculator but rather a statistics displayer. If an OCBPS says that a displayed percent of p% represents the "winning odds", in reality this means not even "this happens in p% from these situations", but rather "this happened in p% of these situations until this moment". â€" Another practical contradiction is in the numerical difference between the real probability of the event and the partial statistic (the relative frequency). â€" It is well known that, as the number of experiments grows, relative frequency approximates the probability (as the limit of the sequence) with higher accuracy, provided that the experiments are performed under identical conditions. In practice, this condition reverts to the fact that the simulations are totally random. We can state with certainty that the external entries are totally random. The internal entries have a high degree of randomness, but this randomness is not total. Mathematics can prove that. (You can read more on this subject in the section titled Relativity of probability in the book Understanding and Calculating the Odds.) The conclusion is that any random generating software cannot create a totally random sequence because of the experimental action. Coming back to OCBPSs, in the light of these facts, we can state that any sequence of simulations pregenerated by an OCBPS does not submit to the conditions of the Law of Large Numbers: the experiments are independent but they are not performed under identical conditions. As to the total number of simulations, to reach a good approximation of the real probability, this number must be huge; and most public odds calculators do not obey this condition, thus the number is otherwise unverifiable. Even if a software program says that its returns are based on 1 million simulations, this figure is irrelevant because only a small portion of those simulations would match a given situation. In addition, if most of those 1 million simulations were automatically pregenerated, the approximations might be altered by the non-random character of the generating program. Still, there are OCBPSs that return very good approximations of the real probability for certain gaming situations. These programs work best for games with very large audiences, like Texas Hold'em and Blackjack. Owing to the high number of external users, such calculators have already cached a huge number of simulations, which helps ensure a satisfactory approximation of the real odds. Still, in these programs this does not happen for every gaming situation or every event within a respective game. More often, they are confined to those frequently inputted by the users, usually for configurations from the first part of the game. In fact, what an OCBPS substitutes through these partial simulations is a complete probability calculus. For some games, this calculus is hard and complex, but for others it is relatively easy. What does a compact formula mean A compact formula is a real function given by a unique algebraic expression that holds one or more variables. Thus, a compact formula is not given in a recurring or iterative mode, it is merely a block expression given in an explicit mode. Any probability calculus involving categories of situations aims to find a compact formula that returns the probability of any event after inputting the parameters that describe the concrete given situation. Thus, to use a compact probability formula means compressing a very large number of concrete situations by extracting from them only the parameters that represent the formula's variables. Any gambling probability application can be solved analytically by using the results of probability theory in a finite sample space. Some of them are complicated and require a lot of time to solve completely, especially for card games. But, as we said, a team of mathematicians and a computer packed with mathematical software can solve any probability aspect of a game in a reasonable time. Why developers choose the partial simulations Obviously, it is far easier to write a code for a software program that performs statistical records of simulations than to solve all the probability formulas of a respective game and then implement them. This latter option assumes the existence of a team of mathematicians to do all the math in a reasonable time, then of a qualified programmer to implement the formulas correctly in the program. Therefore, an OCBPS can be developed by programmers, while a software program based on compact formulas requires one or more mathematicians. Infarom Publishing developed in 2006 and launched in 2007 the first odds calculator entirely based on compact formulas. Called the Draw Poker Odds Calculator, it was built as result of solving all the probability formulas for this variation of poker. We hope this example is followed by other developers because there is still a gap in the market for professional (mathematical) odds calculators. These real odds calculators would enable more players to refer to real odds and not to past statistics in their strategies. Technically, I have nothing against OCBPSs. There is nothing wrong in their way of functioning and their returns are correct. What I contest is the way in which returned information is presented, which misleads users, and some of the descriptions made by their developers. The purpose of this article is to make a comparison from both a mathematical and a practical point of view between the returns of partial simulations and the returns of the compact probability formulas. Many times, the inadvertence is only conceptual and hints of rigorousness. The target practical aspect is a good approximation of the real probability and is accomplished in many cases. However, if we want to use mathematics in gaming strategies or just to provide pure information, we must preserve its character of rigorousness. In fact, games of chance stand as a perfect base for the application of probability theory, which thus far is the only consistent theory that models the hazard with the goal of providing a mathematical measure of the uncertainty. While this measure is the only objective one we have, we must not distort it in any way. Catalin Barboianu is an applied mathematician and author of the books Understanding and Calculating the Odds, Probability Guide to Gambling, Texas Hold'em Odds and Draw Poker Odds. He maintains a probability website with gambling applications at http://probability.infarom.ro . 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