Stochastic and Non-Stochastic Outcomes

Random v. Determined

To this point, the discussion of a coin toss, a roulette spin or a dice throw has presumed one important characteristic: the coin, wheel or die was “fair.” Apart from the ethical questions arising out of cheating , the existence of a non-random influence can distort or even nullify the distribution.
To return briefly to the sixth graders, the results might mislead a researcher if something systematically were explaining height variation, such as gender or age. While there would still be random variation around the mean, the position of the mean and the shape of the distribution (determined by the variance) may look much different from the shape if the non-random elements were taken out of the analysis.

This conclusion is particularly stark in the event of an “unfair” die or wheel or deck of cards. For example, if a die had iron on one side and there were a strong magnet under the table, then “all bets are off.” Literally. The principles governing this branch of probability and statistics apply only to distributions of random variables. When the process is determi-native, as in a crooked die, then there is no real sense in expressing a physical expectation other than whatever the crooked die is programmed to produce.

Or, in the case of the roulette wheel, the analysis was premised on some chance of variation in the outcomes of the spins. They could be anything from 00 to 36. The process that gen-erates the outcomes is stochastic, as there is a random variation in the results. Suppose the roulette wheel were “adjusted” so that no matter what was played or spun, the ball always landed on number 10. It would be hard to describe these outcomes as a “distribution” or the result as a “random variable” because the outcome was pre-determined always to be a 10. If it were described as a random variable, it would have a mean of 10 and a variance of zero. This is not very useful.

Gaming, at least fair gaming, only deals with stochastic processes, that is, variables subject to random variation, in which the mean and variance of the distributions are meaningful. Obviously, if the result were known in advance, there would be no betting.

Past Events

Curiously, any event that is already in the past is not stochastic. It either is, or it is not. It is (or has been) “determined.” So if I hold out my fist and ask you what the probability is that the coin in my hand is a head, the precise, mathematical answer is, “The probability is ei-ther 1 or 0, and unless you show me, I do not know which it is.” Recall that the process of generating the outcomes is what causes random variation. You can not look at a number and tell whether it was the result of a random process. Once the outcome has been “born” it is not random. It is determined. It is the process that matters. (The purists are not entirely right in saying that only future events have probabilities. One can still use probability the-ory to speculate about a determined or past event, the actual outcome of which is still un-known. This contribution to the body of thought in probability theory comes originally from Thomas Bayes, an 18th century British mathematician).

The idea of a random process is essential to understanding gaming. So let’s look at a couple of real world examples. The processes can be anywhere from pure randomness, like the fair roulette wheel, to nearly deterministic, as if a notoriously weak team were mismatched with an overwhelmingly strong one. In each game, it is crucial to understand how the casino sifts out the deterministic elements to leave just the randomized part to bet on. This also applies to games, like poker, where there are betting adversaries other than the house.

A simple principle can be stated: If one squeezes out (accounts for, accommodates) all of the deterministic aspects of the proposition or game move, then the remaining, randomized element, can be wagered on, in the confidence that one knows the real odds. Confident wagering involves peeling back deterministic layers so that the only uncertainties are from random variation.

Examples

In roulette, this is a simple undertaking. Because the edge is usually negative, it will not come up often in the life of a Type II gambler. There are no deterministic aspects to roulette, unless cheating is going on. The odds are 37:1 (or a derivative, according to the bet being made, like “even”).

Likewise, a slot machine will generate its outcome without regard to who is playing it. Spending all weekend doing research on the internet on the non-stochastic aspects of slot machines will not give a player an edge when she pulls the handle or pushes the button.

In the example of Texas Hold’em, the odds of winning were equated with the odds of the next card’s being a spade. While this is a randomized event, the calculation took into ac-count as many of the deterministic aspects of the betting situation as was possible. Recall the process: One could just say that because 1 out of 4 cards is a spade, the odds are 3:1 against. But in truth, the deck had 34 cards (52 less the 14 cards dealt to players and the 4 on the board) and 4 of the spades were accounted for. So the calculation might have been (13-4)/(52-18) or 26.5% approximately. But that would be to assume none of the other players had a spade as a hole card. So to be picky and technical, an estimate of the number of spades in unseen hole cards is required. Those odds were 1 in 4 at the outset, and there were 14 cards dealt, including your own. That would be 3½ spades, including the two you know about. So the estimate of any card in the deck’s being a spade (including the next card, which is the key one) is 7½ divided by 34 or 22%. Note how the deterministic infor-mation was used to winnow down the area of focus on the stochastic or randomized aspect of the play.

To illustrate further this “winnowing down” process, recall also from the Texas Hold’em example that analysis by the player caused the conclusion that the two adversaries almost certainly had a high pair between them and not more than three of a kind. This simplified the betting analysis. If the earlier rounds had not provided information on the hands of the adversaries through the betting process, then the 22% figure would have to change in order to take into account the odds of a full house or better (on one end of the spectrum) or mas-sive bluffing (on the other).

Finally, take the example of the Browns against the Bears next Sunday in Chicago, where the betting proposition is that the total will go (or not go) over 40. Here it is possible to say that there are few if any randomized elements, at least in theory. The objective is to achieve, through research and analysis, as complete a picture of the match-up of the two teams as possible, leading to near certitude about the conduct of the game.

Random elements would only apply to minor aspects of the conduct of the game, it is hoped, like the coin toss or the attentiveness of a given official, or perhaps event to the per-formance of individual actors. The completion percentages for quarterbacks and ends, the success rates for field goal kickers, and the yards per carry of running backs are all ex-pressed as random variables, to be sure, but most sports analysts believe that there is an underlying, non-stochastic explanation for each outcome making up the “statistic.” Like the poker example, the bettor’s objective is to “winnow down” the risk by accounting for all the non-random, non-stochastic elements.

In short, the gaming activities that do reward skill, experience, research and analysis deal with those non-random, non-stochastic, determinative influences over outcomes. If the ca-sino’s book offers a -110 proposition on total runs in the next Padres game, math will help define when it makes sense to bet and when it makes sense to pass. But knowledge is the objective of the serious, rationalist gamer, especially one who considers it his living.

More about Deterministic Gaming

Recall the consideration of the application of statistics and probability to historical events. (What are the odds that this already-flipped coin in my hand is a head?) Purist statisticians object to stating odds in this case not only because it has happened, but more precisely, because it is unique. It can not be replicated over and over. There is no probability distribu-tion about it, so the concept of “expected value” or mean is nonsense in this context.

It would be like asking “What are the odds that Shakespeare was a woman?” One could say the odds are roughly 1 in 2 because genders are almost evenly divided (though that may not have been true in 1564 when he (she?) was born). Shakespeare’s birth is a unique event, to which odds, properly speaking, cannot be applied.

The same can be said of athletic contests. Unlike roulette wheel spins and coin tosses, specific games are unique events. They can not be played “a large number of times.” Only in the most theoretical of senses can you say that there is a “probability distribution” of the final score of the Sunday Browns at Bears game. This means that the calculation of the bet-tor’s edge has a lot less to do with the laws of chance and a lot more to do with the research of historical fact and its analytical synthesis.

In the same manner, horseracing is an activity governed by knowledge, which is derived from familiarity with all of the elements impinging on performance: the horse (pedigree, race history, health, and personality), the trainer, the jockey, the owner, the track, the competition, and a few more things besides.

Consider this sign. It was put on the wall of a team locker room by a high school football coach:

ANY GIVEN TEAM CAN BEAT ANY OTHER GIVEN TEAM, ON ANY GIVEN DAY

The very definition of a “better team” or “faster horse” is that they are “supposed” to win, but motivation might change things. In sports and race betting it is common to include mo-tivation as part of the analysis. If the “better team” has already clinched a playoff spot, and if the underdog’s playoff chances depend on winning the game, then the underdog may perform better than “expected.” This is calculated into the appraisal of the bettor’s edge against the book. But note! If the sports book is the least bit savvy, it will already have fig-ured motivation into its proposition to begin with!

Thus, gaming events are often only partially the result of randomized or stochastic influ-ences. At one end of the spectrum may reside the roulette wheel. At the other is something like the top cede against the bottom cede in a first round bracket of the NCAA basketball tournament. Like any part of our future, the outcome of the NCAA tournament game is still uncertain, even though it is highly predictable, and the unpredictable influences – including the random variations – may be quite small. Somewhere in the middle of this spectrum is poker, perhaps a little toward the NCAA end, but still very much subject to random influ-ences, like being dealt a pair of aces. “Random influences” is just a high-brow way of say-ing “luck” when applied to the careful player.