In the past two years, if you flip open a newspaper, turn on the television, or click onto a webpage, you will find all kinds of “lottery prediction” programs growing increasingly hot. With the flourishing of the lottery industry, this “discipline” of “lottery prediction” has become lively, and a large group of “probability scientists” have found themselves a livelihood.
Lottery prediction programs on television are generally broadcast right after the lottery drawing, and I, in my idle moments, have even watched a few episodes. The professional “lottery scholars” are quite formidable. They can spend dozens of minutes “analyzing” the numbers from the previous draws, know how to use computers to compare the frequency with which each number appears, and often manage, with exceptional inspiration, to organize all kinds of information—for example, by constructing a marvelous matrix of numbers, cleverly placing the winning numbers from previous draws and marking them with various colors, and then—“Look, here there is a line, and then here is a diagonal line… skip over here, along this direction… see, this number has been very active lately! Also, the following combinations have relatively high frequencies; I recommend paying attention to the following numbers…”
On the internet there also seem to be three-dimensional analytical charts available, for example, “×× Network’s latest 3D ‘Pailiesan’ fixed-position single-number prediction: each issue provides 2 single numbers for each position, and promises: at least 6 successful predictions in one position per month; if 6 are not reached, the service fee will be refunded!” It sounds quite mysterious, and there is even a “promise”—the tone is really not modest. But the monthly fee seems to be over a hundred yuan, so of course I would not try it.
So, are their “promises” trustworthy? If not, how do their “doctrines” differ from science? How can we falsify them?
The discovery of scientific laws is likewise an induction of “regularities” from empirical facts. Hume long ago pointed out that such induction has no reliable logical foundation. Moreover, some sciences—especially the social sciences—also concern probabilistic events and use probability for scientific prediction.
Both are based on organizing information from the past few days and calculating, according to some rule, the likelihood that some situation will arise in the future. What, in essence, is the difference between weather forecasting and lottery prediction?
We know that lottery drawings are random, just like tossing a coin: the probability of heads on each toss is 50%, and the result of the previous toss has absolutely nothing to do with the next one! The question is, how do we prove this?
In probability theory, this is called a “goodness-of-fit test.” The specific procedure is: “(1) first assume that this apparatus has no problem; (2) calculate the average result that a standard machine would produce; (3) compare the deviation between the actual data and the calculated average result; (4) calculate the probability p that the result of a standard apparatus would have the same (or greater) deviation; (5) if p is very small, then reject the initial assumption; otherwise, conclude that there is insufficient evidence to show that this apparatus is biased.”[①]
However, the concrete operation is not easy. Suppose we have a defective coin whose probability of coming up heads is only 47%; how many experiments would we need in order to notice the problem?
If we toss it 100 times, it is hard to detect anything wrong. For a standard coin, the probability that the number of heads is more than 60 or fewer than 40 is 5%. When the actual data of the defective coin falls within this 5% range, we can only just barely begin to suspect that something is wrong with the coin. Yet the probability that this defective coin with a 47% heads rate lands between 40 and 60 heads is 93%. In other words, we almost have no chance of noticing the problem in 100 trials, much less proving that the coin is indeed defective. Even if we conduct 1,000 trials, we would begin to suspect the coin only when the number of heads exceeds 532 or falls below 468, but the probability of this happening is still only 50%. To judge with some reliability that this coin is defective, we probably need at least tens of thousands of trials!
The lottery machines are subjected to this kind of “goodness-of-fit test” when they leave the factory. However, if people do not trust those checks carried out behind the scenes, then looking only at the actual drawings from a few hundred rounds is far from enough to establish randomness.
Besides, even if it passes the “goodness-of-fit test,” that still does not mean that no kind of regularity may be hidden behind the data. For example, the sequence 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, … seems to be randomly distributed—but isn’t that just the decimal part of pi? If we perform a “goodness-of-fit test” on the decimal part of pi, it will fit the requirements of randomness perfectly, yet every digit in every position is determined! So when we face a string of data that appears to be completely random, how can we be sure that it is not the product of some regularity such as “the decimal part of 3π2+5π”?
Perhaps everything in the entire universe was determined from the very beginning, and all randomness is merely our illusion. However, whatever the case may be, at present we have not discovered any regularity in it. We have no way to predict which side of the coin will face up next time; therefore, we can only assume that it truly is random, which makes further discussion easier. But the problem is that some lottery scholars claim that they have already discovered certain regularities!
It seems difficult to thoroughly overturn the beliefs of these “lottery scholars.” But there are also perfectly adequate reasons why we believe that lotteries cannot be predicted.
The key lies in the fact that the scientific theories we already know are sufficient to prove this point: “Lottery drawing data are regular and intelligible; they do not require any additional mystical explanation.” By contrast, the theories proposed by the “lottery scholars” have to resort to mysterious factors that are supported by no science whatsoever and are elusive and unpredictable.
Any convincing scientific account (scientific explanation) should, in principle, include one or more “scientific laws” and proceed through reasonable and effective deduction.
Before discussing why scientific explanations are reliable, let us first step back and ask why scientific laws are reliable?
The production of scientific laws requires induction, and it is often even “incomplete induction”—that is, the method primary school students use when doing “find the pattern and fill in the numbers,” rather than the method in middle-school exercises where one derives k+1 from k!
When we see “2, 4, 6, 8, ___, …,” even the dullest primary school student knows how to do this sort of pattern-finding number-filling question. They will unhesitatingly fill in “10” in the blank; slightly smarter students will induce a more general “rule”—“the nth number is 2n.” Although this problem is too simple, its basic principle is the same as the way scientists induce “natural laws”! That is, they “find patterns,” summarize regularities, and make predictions from a series of finite empirical facts or experimental data.
However, middle-school students no longer do “find the pattern and fill in the numbers” problems, because some of the clever students have already realized that such problems have no definite answer: they have infinitely many answers, and any number can be an answer!
From 2, 4, 6, 8, we can summarize the rule as “the nth number is 2n,” but why not the integer part of 2.2n? Or “3n4-40n3+180n2-303n+162”? To decide that the nth number is 2n is nothing more than a matter of “habit.” We habitually favor the simplest answer, the one easiest to think of, but that need not be the true answer. The fact that Newtonian mechanics was surpassed by relativity proves this.
Even so, the “regularities” discovered by science still seem very reliable, at least in people’s feeling. The factors that make scientific laws seem so reliable are:
1. Simplicity: When the two hypotheses An =2n and An =3n4-40n3+180n2-303n+162 both fit experience, people clearly tend to adopt the former. Of course, this may only be a matter of belief. People do not always prefer simplicity. For example, the long-standing preference for the “circle” from ancient Greece to Copernicus seems to have preferred endlessly adding more and more “wheels” rather than considering other approaches…
2. Universality: In fact, universality is also part of simplicity. For example, although the formulas of relativity may look more “complex” than Newtonian mechanics, relativity has a broader scope of application; the phenomena to which one law applies are more widespread. Explaining the broadest range of phenomena with the fewest things—that is universality.
3. Testability (falsifiability): For example, if we “design” a rule An=2n for 2, 4, 6, 8…, then we can anticipate the appearance of the fifth term. If it is indeed “10,” then the rule “An=2n” receives support, and our confidence that the sixth term will be “12” will also increase somewhat.
These are all important characteristics of scientific theories—though they may be neither sufficient nor necessary.
More importantly, in the modern age, various scientific laws and various scientific theories are organized together in mutual support, forming the activity we can regard as a whole called “science.” Although the theories among the various disciplines of the sciences have not been completely unified (this should be impossible), as a whole, “science” is relatively harmonious. Each set of scientific theories has a series of related theories that support one another, and it very rarely clashes violently with other existing theories. In such circumstances, casually deleting or adding a scientific law is by no means easy.
When some “doctrine” demands that the existing scientific theories be massively overhauled, people naturally tend to think it is false. For example, if a special ability such as “reading characters with the ears” were accepted, a great many existing scientific theories would be forced to undergo revisions. As a scientist with normal reasoning, it is only natural to take a strongly skeptical stance toward such a claim. This is not to say that existing scientific theories refuse revision. In fact, constant self-correction is also one of the important reasons why scientific theories appear reliable. However, when someone proposes to overthrow the existing scientific system, one must be clear-headed: the burden of proof lies with him, not with the mainstream scientific community. In other words, mainstream science has no obligation to prove “why the ears can definitely not read characters”; rather, it is the alternative researcher who needs to produce conclusive evidence to demonstrate that “the ears can indeed read characters.”
Likewise, with regard to lotteries, probability theory has already given a scientific explanation: “Lottery numbers are generated randomly, are unrelated to previous results, and cannot be predicted.” And the actual situation does indeed conform to the scientific account. As for whether there is another hidden set of regularities—why lottery numbers have no connection with the positions of the stars—mainstream science does not need to prove this. If someone insists that lottery numbers are connected to the positions of the stars, then that doctrine must necessarily contain new hypotheses that are revolutionary to the existing scientific system, and therefore the burden of proof rests with the proponent of the hypothesis. Since no “lottery scholar” can produce an entire set of convincing new theories, we have every reason to believe that the scientific explanation is the only reliable one.
By the way: “Why do lottery predictions often seem to work quite well?”—this question can also be explained by probability theory.
To be continued
March 21, 2006
[①] See [English] John Haigh, The Mathematics of Chance, trans. Li Daqiang, Jilin People’s Publishing House, 2001, p. 397
Translated from the Chinese original with AI assistance. The original text is authoritative.
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