Theoretical correction of the MCF
The correction of issue 2, obviously requiring that the player should take randomness to be also disorder, is not only a matter of education. It is very hard to convince someone that a logical and unquestionable effect of the independence of the trials is that a succession of occurrences of the same event (outcome or set of outcomes as event) is physically possible in any length. For that, you have also to convince that person that the actual probability of that succession happening (which might be very low, even close to zero) has nothing to do with the truth of the previous statement, just because probability is a limit that is not influenced by the values of, or the order between, the relative frequencies calculated on finite intervals of trials.
That physical possibility is not inconsistent with the stabilization of relative frequencies either, because both the limit and the stabilization are attained “at infinity.” The difficulty resides mainly in perceiving such potential infinity and the relationship with and the place within that infinity of the gambler’s current and past experience. The difficulty of perception is potentiated by cognitive assets that are tightly related to the biological constitution of the human mind. Let’s take them one at a time:
Potential infinity is hard to perceive, even by mathematicians, and it has posed serious problems in the foundation of various concepts and theories during the history of mathematics. For ordinary people, both actual and potential infinity is hard to perceive or understand, just because all our experiences are finite in number.
The infinite sequence of trials for which LLN only makes sense induces a sort of order for the randomness, but this is an order of “the whole” and not of the parts. The frequency of an event over the (infinite) whole (of trials) equals its probability, says LLN in terms of infinity. This is what makes randomness an order, in a cumulative manner. This statement is actually in terms of limits, as there is no arithmetical thing such as a ratio between something and infinity. The number of occurrences of that event over the (potentially) infinite number of trials is infinite, too. Hence the ratio is actually an infinity divided to another infinity, which makes sense only as a limit, not as an arithmetical operation. LLN says that that limit exists and is equal to the probability of that event.
For instance, in the experiment of coin tossing, for the event ‘occurrence of heads,’ having a probability of 1/2, LLN roughly says that the infinity of the occurrences of heads is twice as low as the whole infinity of the trials.
But what happens in the finite bounded interval of the gambling experience of the player affected by MCF? The answer is: Nothing that has any relation with what we said earlier in terms of infinity, even though that interval is part of the whole experience. This can be fairly “translated” as follows: In that interval, randomness is disordered. Summing up in the briefest possible sentence, in what concerns randomness, convergent infinity is order and finiteness is disorder in whatever length of that finiteness. In mathematical terms, finite sets of terms (of whatever cardinal) of a convergent sequence do not count toward the limit of that sequence – there is no mathematical relation between them except that of inclusion between those sets and the entire sequence. If you add or remove terms in finite number to a convergent sequence, its limit remains the same. Things change only if adding/removing infinities. In gambling terms related to MCF, there is no mathematical reason for a succession of occurrences of a random event not to be of any length, since it does not contradict LLN.
When seeing a long succession of the same outcome over a finite interval of trials, the first impression is that the frequency of that outcome increased “abnormally,” and somehow the relative frequency is “in danger” not to be attained in the forthcoming series of trials. The only “reasonable” prediction for overcoming this “danger” is to expect a different outcome in the next trial or a “decent” number of such outcomes over the next interval of trials. This impression is part of the erroneous perception associated with MCF, consisting of a misperception or lack of perception of potential infinity and an unjustified focus on frequency rather than relative frequency. The latter is explicable through the fact that when recording the outcomes of a game, we count those actual outcomes and not any ratio. However, it is the relative frequency that is subject to LLN, and not the frequency. If considering relative frequency, we may see that a long succession of the same outcome is not actually so long with respect to the variations from probability. Let us take a simple illustrative example:
Imagine we toss a coin and record the outcomes heads (H) or tails (T). Assume the following distribution of outcomes over the first eight trials: T H H H H H H H … The occurrence of seven consecutive heads may trigger for some people the MCF, with the expectation for a T to come up on the next trial. Let us see in numbers whether the seven H justifies the qualification of “abnormal” that I mentioned previously. In the next table are the frequency of Hs and the relative frequency of Hs calculated after each trial.
No. of trial: 1 2 3 4 5 6 7 8
Outcome: T H H H H H H H
Frequency: 0 1 2 3 4 5 6 7
Relative frequency: 0 1/2 2/3 3/4 4/5 5/6 6/7 7/8
(fraction)
Relative frequency: 0 0.5 0.66 0.75 0.8 0.86 0.85 0.87
(decimal fraction)
Over the interval of trials from 2 to 8 there were 7 heads recorded, instead of 3-4, what was expected as “normal,” since the probability for a head is 1/2. A difference of about 3-4 units from the “normal” prior expectation is recorded. But if we look at how the relative frequency of heads changed over that interval, we see it ranging from 0.5 to 0.87, that is, with about 3 to 4 decimals. Moreover, imagine that the new pending outcome is T. At this ninth trial, the relative frequency of the heads is now 7/9 = 0.77. It decreased back 1 decimal for only one trial, coming closer to the probability of 0.5. With these numbers in hand, the “abnormal” behavior of the outcomes does not seem so hard to be restored to “normal” with the forthcoming trials.
The effect is visible also when the interval in question shows many Hs, but also Ts. Assume one T occurs in that interval: T H H T H H H H … The new table is as follows:
No. of trial: 1 2 3 4 5 6 7 8
Outcome: T H H T H H H H
Frequency: 0 1 2 2 3 4 5 6
Relative frequency: 0 1/2 2/3 2/4 3/5 4/6 5/7 6/8
(fraction)
Relative frequency: 0 0.5 0.66 0.5 0.6 0.66 0.71 0.75
(decimal fraction)
Again, there is a frequency difference of about 3 heads from the “normal” over that interval and only of 0.25 for the relative frequency. If a T occurs in the next trial, the relative frequency of heads becomes 0.66, so it decreases back to about 1 decimal for only one trial.
So many times in gambling we tend to measure what happens or what we expect to happen through numbers reflecting our immediate perception – as one would count the occurrences of heads in our previous example (instead of the relative frequency of heads) and make predictions based on such measurements. This behavior of using a wrong or inadequate measure in evaluating a situation with respect to expectations also stands at the origin of other gambling cognitive distortions, as we shall see in a forthcoming chapter.
However, even using the right measure does not always guarantee the elimination of all fallacies. This also applies to MCF. A long succession of the same outcome is – correctly – associated with a very low probability of happening. In our previous example, the probability for occurrence of seven heads in a row (or tails, as the same probability stands), calculated right after the first trial, is (1/2)^7 = 0.0078125 = 0.78%, which a low one. Probabilities for successions longer than 7 are even lower. Still, it happens. The historical occurrence of 26 blacks in a row in 1913 had its probability of (18/37)^26, which is about 0.000000007308, about 137 million to 1, that is, almost zero. Still, the event occurred at some moment among the billions of roulette spins over its history.
We started our discussion on correcting the MCF by saying that a first step is that the player must be convinced about the physical possibility of the occurrences of a succession of unfavorable outcomes of any length, say 1,000, despite its extremely low probability. Even though such an event may not occur during the player’s lifetime or even several centuries after, it will occur at some moment (or it has already occurred in the quantum multiverse, just to make fun of it a bit). There is no preferential period or moment for such occurrence due to the disorder of randomness, and hence it may occur at the moment of the play as well; there is no logical reason or law of nature for negating that possibility.
Of course, being convinced of the physical possibility of a succession of 1,000 reds in roulette must not imply the expectation for 1,000 or so occurrences of black in a near future interval, which balances the opposite series. If that were the case, the initial conviction would be erroneous as well.
Such a conviction is still hard to achieve because many people (including experts) see or perceive that physical possibility as constrained by the LLN in the sense that the convergence stated by LLN, and implicitly the stabilization of the relative frequency, seem to be inconsistent with the independence of the trials. The fallacy of such a premise resides in the conceptual framework, where the concept of independence should be the statistical one, since we reason in mathematical terms. Besides that, it is a matter of perception – an unclear perception either of potential infinity or of how LLN really works affects that conviction.
Coming back to the three educational causes of the MCF I have discussed, their correction counterparts would be these:
1) employing statistical (in)dependence in reasoning instead of physical causal (in)dependence;
2) taking randomness to be disorder;
3) distinguishing between probability and relative frequency, and between frequency and relative frequency.
Most of the arguments for these three directions of education have been detailed previously and are not exclusively mathematical, but also conceptual-epistemological. As I already mentioned, there are also psychological factors that the MCF and its correction depend on. Some of these factors are cognitive and deeply rooted in our psychobiological constitution.
Humans are constructed and equipped by evolution to look for equilibrium and safety. We do not like incertitude, and we tend to evaluate things and situations by basing on indisputable, real facts. This may explain why so many people rely on recorded statistics and frequencies rather than on mathematical measures like probability – what has happened in the past is certifiable fact, which probability seems not to be. The human brain always looks for equilibrium, too. The human brain is a huge consumer of energy, and since energy-saving is an evolutionary feature of the living organisms, the brain has developed various physiological ways to save energy. We always look for causes as associations for facts that seem to have jumped out of the usual patterns of our experience and universe of beliefs, in order to reach a mental state of equilibrium. This habit is so strong that sometimes we associate causes with a fact or event when we should not do that, leaving it unexplained. It is so strong that we may even come to no longer believe in the independence of the trials of a random experiment, just to “explain” something that we cannot explain otherwise.
A good suggestion for those affected by MCF who exhibit causes 1 and/or 2 related to independence and randomness is a mental exercise requiring the player to imagine each new trial as if it was the first, and consequently, to ignore the previous outcomes. This is supposed to solve the issue; however, whether the subject is able to achieve that depends not only on the knowledge acquired through education or the way it is delivered, since the causes also have a psychological and even a neurological component. Among other cognitive factors assumed to prevent the correction of the MCF through education, there are cognitive bias, superstitions, supernatural/paranormal beliefs, beliefs in fate/destiny, etc. Let us note that such factors may also count as causes of the MCF.