We have found so far the most general consistent rules by which our robot can manipulate plausibilities, granted that it must associate them with real numbers, so that its brain can operate by the carrying out of some definite physical process. While we are encouraged by the familiar formal appearance of these rules and their qualitative properties just noted, two evident circumstances show that our job of designing the robot’s brain is not yet finished.

到目前为止,我们已经找到了机器人处理似真度的最一般的一致规则,即必须将似真度关联到一个实数,然后机器人的大脑才能以某种明确的物理过程来完成运作。虽然这些规则的形式看起来很熟悉而且其定性属性让我们感到鼓舞,但两个明显的情况表明我们设计机器人大脑的工作还没有完成。

In the first place, while the rules (2.63), (2.64) place some limitations on how plausibilities of different propositions must be related to each other, it would appear that we have not yet found any unique rules, but rather an infinite number of possible rules by which our robot can do plausible reasoning. Corresponding to every different choice of a monotonic function p(x), there seems to be a different set of rules, with different content.

首先,虽然公式(2.63)和(2.64)给出了对不同命题的似真度之间的关系的一些限制,但似乎似真推断的规则的数目并不唯一而是无穷个。对应于选中的每个单调函数p(x),似乎存在不同的规则集,具有不同的内容。

Secondly, nothing given so far tells us what actual numerical values of plausibility should be assigned at the beginning of a problem, so that the robot can get started on its calculations. How is the robot to make its initial encoding of the background information into definite numerical values of plausibilities? For this we must invoke the ‘interface’ desiderata (IIIb), (IIIc) of (1.39), not yet used.

其次,到目前为止,我们并不知道在开始处理问题时,应该赋予似真度什么数值,以便机器人可以开始工作。机器人如何为背景信息确定初始似真度的数值？为此,我们必须引用尚未使用的'界面'基础原理,即(1.39)中的(IIIb)和(IIIc)。

The following analysis answers both of these questions, in a way both interesting and unexpected. Let us ask for the plausibility $(A_1+A_2+A_3|B)$ that at least one of three propositions $\{A_1,A_2,A_3\}$ is true.We can find this by two applications of the extended sum rule (2.66), as follows. The first application gives

以下分析以既有趣又意外的方式回答了这些问题。 考虑$(A_1+A_2+A_3|B)$的似真度,其中三个命题$\{A_1,A_2,A_3\}$中至少一个为真。我们可以如下应用两次扩展和规则(2.66)。 第一次应用给出

where we first considered $(A_1+A_2)$ as a single proposition, and used the logical relation

首先将$(A_1+A_2)$看作单个命题,并使用逻辑关系

Applying (2.66) again, we obtain seven terms which can be grouped as follows:

再次应用(2.66),得到如下分组的7项

Now suppose these propositions are mutually exclusive; i.e. the evidence B implies that no two of them can be true simultaneously:

现在假设这些命题彼此独立,例如证据B蕴含了这些命题没有任何两个可以同时为真

Then the last four terms of (2.82) vanish, and we have

然后(2.82)的最后4项消除了,有

$p(A_1+A_2+A_3|B)=p(A_1|B)+P(A_2|B)+P_(A_3|B)$ (2.84)

Adding more propositions $A_4, A_5$, etc., it is easy to show by induction that if we have n mutually exclusive propositions $\{A_1, ... , A_n\}$, (2.84) generalizes to

增加更多命题如$A_4, A_5$等等,容易归纳出,如果有n个彼此独立的命题$\{A_1, ... , A_n\}$,(2.84)可推广为

a rule which we will be using constantly from now on.

这个规则从现在开始我们会经常用到.

In conventional expositions, Eq. (2.85) is usually introduced first as the basic but, as far as one can see, arbitrary axiom of the theory. The present approach shows that this rule is deducible from simple qualitative conditions of consistency. The viewpoint which sees (2.85) as the primitive, fundamental relation is one which we are particularly anxious to avoid (see Comments section at the end of this chapter).

在传统的论述中,等式(2.85)通常作为最基本的东西被首先介绍,但如你所见,作为一个"天选"的公理。我们的方法表明,该规则可以从定性这个简单的定性条件中推导出来。 将(2.85)视为原始且基本的关系的观点,是我们特别想避免的观点(参见本章末尾的评论部分)。

Now suppose that the propositions $\{A_1, ... , A_n\}$ are not only mutually exclusive but also exhaustive; i.e. the background information B stipulates that one and only one of them must be true. In that case, the sum (2.85) for m = n must be unity:

现在假设命题$\{A_1,...,A_n\}$不仅互相排斥,而且是穷尽的;即,背景信息B规定其中一个且仅一个必须为真。 在这种情况下,当m=n时和(2.85)必须是1：

This alone is not enough to determine the individual numerical values $p(A_i|B)$. Depending on further details of the information B, many different choices might be appropriate, and in general finding the $p(A_i|B)$ by logical analysis of B can be a difficult problem. It is, in fact, an open-ended problem, since there is no end to the variety of complicated information that might be contained in B; and therefore no end to the complicated mathematical problems of translating that information into numerical values of $p(A_i|B)$. As we shall see, this is one of the most important current research problems; every new principle we can discover for translating information B into numerical values of $p(A_i|B)$ will open up a new class of useful applications of this theory.

仅此还不足以确定所有的单个数值$p(A_i|B)$。根据信息B的进一步细节,可能有许多不同的合理选择,并且一般来说通过对B的逻辑分析来找到所有的$p(A_i|B)$可能是非常难题的。事实上这是一个开放的问题,因为B中可能包含的各种复杂信息是无止境的;因此将这些信息转换为对应的$p(A_i|B)$的数值,也是个无止境的复杂数学问题。正如我们将要看到的,这是当前最重要的研究问题之一;我们可以发现将信息B转换为对应所有$p(A_i| B)$的数值的每一个新原理,都将开辟这一理论的一类新的有意义的应用。

There is, however, one case in which the answer is particularly simple, requiring only direct application of principles already given. But we are entering now into a very delicate area, a cause of confusion and controversy for over a century. In the early stages of this theory, as in elementary geometry, our intuition runs so far ahead of logical analysis that the point of the logical analysis is often missed. The trouble is that intuition leads us to the same final conclusions far more quickly, but without any correct appreciation of their range of validity. The result has been that the development of this theory has been retarded for some 150 years because various workers have insisted on debating these issues on the basis, not of demonstrative arguments, but of their conflicting intuitions.

然而,在一些情况下,答案是如此简单,只需要直接应用已有的原则就够了。但是,我们现在要处理的问题是如此微妙,以至于导致了一个多世纪的混乱和争议。在这个理论的早期阶段,就像在初等几何学中一样,我们的直觉远远的跑在了逻辑分析之前,以至于经常丢弃了逻辑分析。问题在于直觉很快就让我们得出了和逻辑分析相同的最终结论,但却没有了解结论合法的边界。结果是这个理论的发展已经被推迟了大约150年,因为人们都坚持在他们互相冲突的直觉的基础上讨论问题,而不是基于展示各自的论证过程。

At this point, therefore, we must ask the reader to suppress all intuitive feelings you may have, and allow yourself to be guided solely by the following logical analysis. The point we are about to make cannot be developed too carefully; and, unless it is clearly understood, we will be faced with tremendous conceptual difficulties from here on. Consider two different problems. Problem I is the one just formulated: we have a given set of mutually exclusive and exhaustive propositions $\{A_1, ... , A_n\}$ and we seek to evaluate $p(A_i|B)_I$ . Problem II differs in that the labels $A_1, A_2$ of the first two propositions have been interchanged. These labels are, of course, entirely arbitrary; it makes no difference which proposition we choose to call $A_1$ and which $A_2$. In Problem II, therefore, we also have a set of mutually exclusive and exhaustive propositions $\{A'_1, , A'_n\}$, given by

因此,在这一点上,我们必须要求读者放弃所有的直觉,并且完全遵从下面的逻辑分析。我们即将制定的观点不能过于谨慎;而且,除非清楚地理解,否则我们将面临巨大的概念上的困难。考虑两个不同的问题。问题I阐述为：有一套相互排斥且穷尽的命题集$\{A_1,...,A_n\}$,我们试图评估$p(A_i|B)_I$。问题II的不同之处在于前两个命题的标签$A_1,A_2$被互换了。当然,这些标签完全是任意的;我们选择将哪个命题称为$A_1$以及哪个命题为$A_2$没有任何实质区别。因此,在问题II中,我们也有一组相互排斥且穷尽的命题集$\{A'_1,,A'_n\}$,如下

and we seek to evaluate the quantities $p(A'_i|B)_{II}$ , i = 1, 2, …​ , n.

我们试图评估$p(A'_i|B)_{II}$的具体数值,其中i = 1, 2, …​ , n.

In interchanging the labels, we have generated a different but closely related problem. It is clear that, whatever state of knowledge the robot had about A1 in Problem I, it must have the same state of knowledge about $A'_2$ in Problem II, for they are the same proposition, the given information B is the same in both problems, and it is contemplating the same totality of propositions $\{A_1, ... , A_n\}$ in both problems. Therefore we must have

通过交换标签,我们得到了一个不同但类似的问题。 显然无论机器人在问题I中对A1有什么知识状态,它在问题II中必须具有相同的关于$A'_2$的知识状态,因为它们本就是相同的命题,而且两个问题都是在给定的信息B的前提下,并且在两个问题中都考虑了相同的命题集$\{A_1,...,A_n\}$。 因此我们必须有

and similarly

类似的

We will call these the transformation equations. They describe only how the two problems are related to each other, and therefore they must hold whatever the information B might be; in particular, however plausible or implausible the propositions $A_1, A_2$ might seem to the robot in Problem I.

我们将这些称为变换方程。它们只描述了这两个问题是如何相互关联的,因此它们必须和B保持一致无论B包含了什么信息;特别是,机器人认为在问题I中$A_1,A_2$有多可信或多不可信。

Now suppose that information B is indifferent between propositions $A_1$ and $A_2$; i.e. if it says something about one, it says the same thing about the other, and so it contains nothing that would give the robot any reason to prefer either one over the other. In this case, Problems I and II are not merely related, but entirely equivalent; i.e. the robot is in exactly the same state of knowledge about the set of propositions $\{A'_1, ... , A'_n\}$ in Problem II, including their labeling, as it is about the set $\{A_1, ... , A'_n\}$ in Problem I.

现在假设信息B在命题$A_1$和$A_2$之间并无不同;也就是说,如果它对一个命题陈述了什么,那么它对另一个命题陈述了相同的东西,因此它没有包含任何信息可以让机器人有任何理由偏爱任何一个而不是另一个。在这种情况下,问题I和II不仅仅是相关的,而是完全相同的;即机器人在问题II中的命题集${\A'_1,...,A'_n\}$所处于的知识状态,包括命题的标签,与问题I的命题集$\{A_1,…​,A'n\}$完全一样. Now we invoke our desideratum of consistency in the sense (IIIc) in (1.39). This stated that equivalent states of knowledge must be represented by equivalent plausibility assignments. In equations, this statement is 现在我们引用(1.39)中的一致性基础公理(IIIc)。该公理声明,同等的知识状态必须用相等的似真度数值来表示。在方程中,这个陈述是$p(A_i|B)_I=p(A'_i|B){II}$, i = 1, 2, . . . , n, (2.90) which we shall call the symmetry equations. But now, combining (2.88), (2.89), and (2.90), we obtain 我们称此为对称方程.现在,结合(2.88),(2.89)和(2.90),我们得到$p(A_1|B)I=p(A_2|B)_I$. (2.91) In other words, propositions A1 and A2 must be assigned equal plausibilities in Problem I (and, of course, also in Problem II). 换句话说,在问题I中,命题A1和A2必须被赋予相同的似真度(当然,在问题II中也相同)。 At this point, depending on your personality and background in this subject, you will be either greatly impressed or greatly disappointed by the result (2.91). The argument we have just given is the first ‘baby’ version of the group invariance principle for assigning plausibilities; it will be extended greatly in Chapter 6, when we consider the general problem of assigning ‘noninformative priors’. 到这里为止,依赖你关于这个主题拥有的背景和偏好,你可能会对这个结果(2.91)感到印象深刻或非常失望(2.91)。 我们刚才给出的论点是对似真度进行赋值的群体不变性原则的第一个“婴儿”版本;在第六章中将对此进行深度扩展,当我们考虑如何分配“非信息性的先验信息”的一般问题时。 More generally, let$\{A''_1, , A''_n\}$be any permutation of${A_1, , A_n\}$and let Problem III be that of determining the$p(A''_i|B)$. If the permutation is such that$A''_k≡A_i$, there will be n transformation equations of the form 更一般地说,让$\{A'' 1,A''_ n\}$为${A_1,,A_n\}$的任意排列,让问题III为确定$p(A''I|B)$。 如果排列是$A''_k≡A_i$,那么将有n个变换方程形式为$p(A_i|B)_I=p(A''_k|B){III}$(2.92) which show how Problems I and III are related to each other; these relations will hold whatever the given information B. 这表明问题I和III是如何相互关联的; 这些关系将和给定的信息B保持一致. But if information B is now indifferent between all the propositions Ai , then the robot is in exactly the same state of knowledge about the set of propositions$\{A''1, …​ , A''_n\}$in Problem III as it was about the set${A_1, , A_n\}$in Problem I; and again our desideratum of consistency demands that it assign equivalent plausibilities in equivalent states of knowledge, leading to the n symmetry conditions 但是,如果信息B现在对所有命题Ai都没有差别,那么机器人对问题III中的命题集$\{A'' 1,…​,A''n\}$的知识状态,与问题I中的命题集$\{A_1,,A_n\}$完全一样; 再一次,我们的一致性基础公理要求对同等的知识状态赋予相等的似真度,从而导致n个对称条件$p(A_k|B)_I=p(A''_k|B){III}$, k = 1, 2, . . . , n. (2.93) From (2.92) and (2.93) we obtain n equations of the form 从(2.92)和(2.93)我们得到n个方程,形式为$p(A_i|B)_I=p(A_k|B)_I$. (2.94) Now, these relations must hold whatever the particular permutation we used to define Problem III. There are n! such permutations, and so there are actually n! equivalent problems among which, for given i , the index k will range over all of the (n − 1) others in (2.94). Therefore, the only possibility is that all of the$p(A_i|B)_I$be equal (indeed, this is required already by consideration of a single permutation if it is cyclic of order n). Since the${A_1, , A_n\}$are exhaustive, (2.86) will hold, and the only possibility is therefore 现在,这些关系必须成立,无论我们用于定义问题III的任何排列。一共有n！这样的排列,所以实际上有n！个等价的问题,其中对于给定的第i个问题,下标k将在(2.94)中的所有其他(n-1)个范围内。 因此,唯一可能的就是所有$p(A_i|B)_I$都是相等的(实际上,只考虑一个排列的同样如此,如果它按n循环)。 由于${A_1,,A_n\}$是穷尽的,(2.86)仍将成立,因此唯一的可能是$p(A_i|B)_I= \frac 1 n$, (1 ≤ i ≤ n), (2.95) and we have finally arrived at a set of definite numerical values! Following Keynes (1921), we shall call this result the principle of indifference. 我们终于得出了一组确定的数值！ 在凯恩斯(1921)之后,我们将这个结果称为无差别原则。 Perhaps, in spite of our admonitions, the reader’s intuition had already led to just this conclusion, without any need for the rather tortuous reasoning we have just been through. If so, then at least that intuition is consistent with our desiderata. But merely writing down (2.95) intuitively gives one no appreciation of the importance and uniqueness of this result. To see the uniqueness, note that if the robot were to assign any values different from (2.95), then by a mere permutation of labels we could exhibit a second problem in which the robot’s state of knowledge is the same, but in which it is assigning different plausibilities. 也许,读者不需要我们的指示而凭直觉就能得出这个结论,不需要我们刚刚经历过的相当曲折的推理。如果是这样,那么至少这种直觉是与我们的基础公理一致的。但仅是凭直觉而写出(2.95),就不会让人认识到这个结果的重要性和唯一性。要看到唯一性,请注意,如果机器人分配的任何与(2.95)不同的数值,那么仅通过标签的私信排列,我们就找到又一个问题,虽然机器人对两者拥有相同的知识状态,但分配的似真度却是不同的。 To see the importance, note that (2.95) actually answers both of the questions posed at the beginning of this section. It shows – in one particular case which can be greatly generalized – how the information given the robot can lead to definite numerical values, so that a calculation can start. But it also shows something even more important because it is not at all obvious intuitively; the information given the robot determines the numerical values of the quantities$p(x)=p(A_i|B)$, and not the numerical values of the plausibilities$x=A_i|B$from which we started. This, also, will be found to be true in general 重要的是,请注意(2.95)实际上回答了本节开头提出的两个问题。它表明 - 一个特殊情况亦是通常情况下 - 给予机器人的信息如何能够产生确定的数值,从而作为计算的起点。但它也显示出更为重要的东西,因为它本非直观;给定机器人的信息决定了我们的出发点,即$p(x)=p(A_i|B)$的数值,而不是似真度$x=A_i|B$的值。一般来说,这也是正确的。 Recognizing this gives us a beautiful answer to the first question posed at the beginning of this section; after having found the product and sum rules, it still appeared that we had not found any unique rules of reasoning, because every different choice of a monotonic function p(x) would lead to a different set of rules (i.e. a set with different content). But now we see that no matter what function p(x) we choose, we shall be led to the same result (2.95), and the same numerical value of p. Furthermore, the robot’s reasoning processes can be carried out entirely by manipulation of the quantities p, as the product and sum rules show; and the robot’s final conclusions can be stated equally well in terms of the p’s instead of the x’s. 认识到这一点为我们提供了对本节开头提出的第一个问题的美妙答案;在找到乘积和加法规则之后,我们似乎还没有找到推理的唯一规则,因为选择任何一个单调函数p(x)都会导致一组不同的规则(即一组不同的内容)。但是现在我们看到无论我们选择什么函数p(x),我们都会得到相同的结果(2.95)和p的相同数值。之后,机器人的推理过程可以完全通过操作数量p来执行,如乘法和加法规则所示;并且机器人的最终结论可以等价的用p替换x来示。 So, we now see that different choices of the function p(x) correspond only to different ways we could design the robot’s internal memory circuits. For each proposition$A_i$about which it is to reason, it will need a memory address in which it stores some number representing the degree of plausibility of$A_i$, on the basis of all the data it has been given. Of course, instead of storing the number$p_i$it could equally well store any strict monotonic function of$p_i$. But no matter what function it used internally, the externally observable behavior of the robot would be just the same. 因此,我们现在看到函数p(x)的不同选择仅对应于我们设计机器人内部存储器电路的不同方式。 对于每个被推理的命题$A_i$,它将需要一个存储器地址来存储代表$A_i$的似真度的数字,基于给定的所有数据。 当然,它可以同样存储$p_i$的任何严格的单调函数,而不是数字$p_i$。但无论机器人内部使用什么机制,其外部可观察的行为都是相同的。 As soon as we recognize this, it is clear that, instead of saying that p(x) is an arbitrary monotonic function of x, it is much more to the point to turn this around and say that: 一旦我们认识到这一点,很明显,与其说p(x)是x的任意单调函数,不如说： The plausibility x ≡ A|B is an arbitrary monotonic function of p, defined in (0 ≤ p ≤ 1). x≡A|B的似真度是p的任意单调函数,0 ≤ p ≤ 1 It is p that is rigidly fixed by the data, not x. 数据严格决定了p,而不是x. The question of uniqueness is therefore disposed of automatically by the result (2.95); in spite of first appearances, there is actually only one consistent set of rules by which our robot can do plausible reasoning, and, for all practical purposes, the plausibilities x ≡ A|B from which we started have faded entirely out of the picture! We will just have no further use for them. 因此,结果(2.95)自动保证了唯一性;尽管第一次出现,实际上只有一套一致的规则,我们的机器人可以通过这些规则进行似真推理,并且,出于所有实际目的,起始的x≡A|B的似真度已经完全消失在视野之外了！我们将再也不会用到它们。 Having seen that our theory of plausible reasoning can be carried out entirely in terms of the quantities p, we finally introduce their technical names; from now on, we will call these quantities probabilities. The word ‘probability’ has been studiously avoided up to this point, because, whereas the word does have a colloquial meaning to the proverbial ‘man on the street’, it is for us a technical term, which ought to have a precise meaning. But until it had been demonstrated that these quantities are uniquely determined by the data of a problem, we had no grounds for supposing that the quantities p were possessed of any precise meaning. 看到我们的似真推理理论完全仅可以基于操作数量p进行,我们现在终于可以引入其技术性的名称;从现在开始,我们将称这些数量为概率。之前我们刻意避免使用“概率”这个词,因为这个词有非正式的“大部分人的看法”的意味,但对一个技术术语来说应该具有精确的含义。但是,在证明这些数量是由问题的信息唯一确定之前,我们无法证明数量p是具有某种确切含义的。 We now see that the quantities p define a particular scale on which degrees of plausibility can be measured. Out of all possible monotonic functions which could, in principle, serve this purpose equally well, we choose this particular one, not because it is more ‘correct’, but because it is more convenient; i.e. it is the quantities p that obey the simplest rules of combination, the product and sum rules. Because of this, numerical values of p are directly determined by our information. 我们现在看到数量p定义了一个似真度的度量方式。在所有可行的单调函数中,原则上是同样有效的,但我们选择一个个特定的函数,不是因为它更“正确”,而是因为它更方便;即,遵守最简单的组合规则,乘法规则和加法规则的度量p。因此,p的数值直接由我们的信息确定。 This situation is analogous to that in thermodynamics, where out of all possible empirical temperature scales t, which are monotonic functions of each other, we finally decide to use the Kelvin scale T ; not because it is more ‘correct’ than others but because it is more convenient; i.e. the laws of thermodynamics take their simplest form [dU = T dS − PdV, dG = −SdT + VdP, etc.] in terms of this particular scale. Because of this, numerical values of temperatures on the kelvin scale are ‘rigidly fixed’ in the sense of being directly measurable in experiments, independently of the properties of any particular substance like water or mercury. 这和热力学很像,在所有可能的经验性的温标t中,都是单调函数,但我们最终决定使用开尔文温标T;不是因为它比其他选择更“正确”,而是因为它用起来更方便.热力学定律在使用开式温标时,公式的形式最简单,例如[dU=TdS-PdV,dG=-SdT+VdP]等等。因此,在实验中可直接测量的意义上,开尔文尺度上的温度数值是“严格固定的”,与任何特定物质如水或汞的性质无关。 Another rule, equally appealing to our intuition, follows at once from (2.95). Consider the traditional ‘Bernoulli urn’ of probability theory; ours is known to contain ten balls of identical size and weight, labeled {1, 2, . . . , 10}. Three balls (numbers 4, 6, 7) are black, the other seven are white. We are to shake the urn and draw one ball blindfolded. The background information B in (2.95) consists of the statements in the last two sentences. What is the probability that we draw a black one? 另一条同样符合直觉的规则,可以立刻从(2.95)导出。考虑概率论中传统的“伯努利实验”,已知盒子里有十个大小重量完全相同的球,标记为{1,2,....,10}。三个球(标记数字4,6,7)是黑色的,另外七个是白色的。我们摇动盒子蒙上眼睛取出一个球。(2.95)中的背景信息B由上面最后两句的陈述组成。我们取出一个黑球的概率是多少？ Define the propositions:$A_i≡$‘the ith ball is drawn’, (1 ≤ i ≤ 10). Since the background information is indifferent to these ten possibilities, (2.95) applies, and the robot assigns 定义命题：$A_i≡$取出的第i个球,(1≤i≤10)。由于背景信息对十种可能并无差别,那么可以应用(2.95),机器人应分配$p(A_i|B)= \frac 1 10$, 1 ≤ i ≤ 10. (2.96) The statement that we draw a black ball is that we draw number 4, 6, or 7; 取出一个黑球,即取出标号为4,6或7的球$p(black|B)=p(A_4+A_6+A_7|B)$. (2.97) But these are mutually exclusive propositions (i.e. they assert mutually exclusive events), so (2.85) applies, and the robot’s conclusion is 但这些是彼此互斥的命题(即,是对等且不相容的事件),所以应用(2.85)机器人可以得出结论$p(black|B) = \frac 3 10$, (2.98) as intuition had told us already. More generally, if there are N such balls, and the proposition A is defined to be true on any specified subset of M of them, (0 ≤ M ≤ N), false on the rest, we have 和我们的直觉一样.更普遍的,如果有N个球,命题A为真表示取出的是子集M中的一个,(0 ≤ M ≤ N),否则为假,那么有$p(A|B) = \frac M N$$. (2.99)

This was the original mathematical definition of probability, as given by James Bernoulli (1713) and used by most writers for the next 150 years. For example, Laplace’s great Th´eorie Analytique des Probabilit´es (1812) opens with this sentence:

这是詹姆斯伯努利(James Bernoulli,1713)给出的概率的原始数学定义,并被大多数作家接下来使用了150年。例如,拉普拉斯伟大的著作"Th’eorie Analytique des Probabilit’es"(1812)以下面这句话开始：

The Probability for an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.

一个事件的概率是有利于它的事件的数量与所有事件的数量之比,当没有任何理由使得我们预期某些事件应该比其他事件发生更多次时,对我们来说这意味着其概率是相等的。