The relativity of space and time implies thattime measurements have also a minimum error, given in fact by
Δt≥ℏ/2ΔE,
where ΔE is the error in our knowledge of the energy of the process whose timeperiod we are measuring. If we wish to know more precisely whensomething happened we must know less about what happened, because ourknowledge of the energy involved will be less. The time uncertainty is alsorelated to the wave nature of matter.
空间和时间的相对性，也暗示着，时间测量，也有一个最小的错误，通过下面的事实给出：
Δt≥ℏ/2ΔE,
这里ΔE，是我们关于过程的能量的知识中的错误，我们正在测量的，就是这个过程的时间周期。当某事发生的时候，如果我们{对时间}希望知道地更精确，那么，对于究竟发生了什么，我们就应该知道的要更少，因为我们关于所牵扯到的能量的知识，将会更少。时间的不确定性，同样与物质的波的的特性有关。

1. This equation is right only if the area covered by the nuclei is asmall fraction of the total, i.e., if (n1−n2)/n1 is much less than 1 . Otherwise we must make a correction for the fact that some nucleiwill be partly obscured by the nuclei in front of them.
脚注1、只有当原子核所占据的面积，是总面积的一个小的部分时，亦即，当(n1−n2)/n1，远远地小于1时，这个公式才是正确的。否则的话呢，我们就应该为下面的事实，做一个更正，该事实就是：有些原子核，被它前面的原子核给遮住了，从而部分地变模糊了。{部分地重叠了}

6–1Chance and likelihood 6-1机会与可能性
“Chance” is a word which is in common usein everyday living. The radio reports speaking of tomorrow’s weather may say:“There is a sixty percent chance of rain.” You might say: “There is a smallchance that I shall live to be one hundred years old.” Scientists also use theword chance. A seismologist may be interested in the question: “What is thechance that there will be an earthquake of a certain size in Southern Californianext year?” A physicist might ask the question: “What is the chance that aparticular geiger counter will register twenty counts in the next ten seconds?”A politician or statesman might be interested in the question: “What is thechance that there will be a nuclear war within the next ten years?” You may beinterested in the chance that you will learn something from this chapter.
“机会”这个词，在日常生活很常见。广播在谈到明天的天气时，会说："降雨的机会大概是60%"。你也可能会说：“我活到100岁的机会是多大呢？”一个地震学家没准会对下述问题感兴趣：“明年在加利福尼亚的南部，发生一定程度地震的机会是多少？”一个物理学家可能会问这样的问题：“在下一个十秒中，一个盖革计数器，计数到20的机会是多少？”一个政客或政治家，或许对下面的问题有兴趣：“在下一个十年中，爆发一场核战的机会有多大？”你从这一章，或许能学到点什么，对此机会，你可能会感兴趣。

By chance, we mean something like aguess. Why do we make guesses? We make guesses when we wish to make a judgmentbut have incomplete information or uncertain knowledge. We want to make a guessas to what things are, or what things are likely to happen. Often we wish tomake a guess because we have to make a decision. For example: Shall I take myraincoat with me tomorrow? For what earth movement should I design a newbuilding? Shall I build myself a fallout shelter? Shall I change my stand ininternational negotiations? Shall I go to class today?
通过机会这个词，我们所意味的，就是猜测。我们为什么要猜呢？我们之所以要猜，乃是因为，我们希望做一个判断，但是，拥有的信息不够、知识不确定。所以我们就想猜测，事情是什么样子的，或什么事情将会发生。通常，我们希望做一个猜测，那是因为我们必须做一个决定。例如，我明天是否应该带上雨衣？为了某种地球的运动，我应该设计出什么样的新建筑？我是否应该给我自己建一个核辐射掩蔽所？在国际谈判中，我是否应该改变立场？今天我是否应该去上课？

Sometimes we make guesses because we wish,with our limited knowledge, to say as much as we can about somesituation. Really, any generalization is in the nature of a guess. Any physicaltheory is a kind of guesswork. There are good guesses and there are badguesses. The theory of probability is a system for making better guesses. Thelanguage of probability allows us to speak quantitatively about some situationwhich may be highly variable, but which does have some consistent averagebehavior.
我们有时候猜测，乃是因为我们希望用我们有限的知识，对某些情况，尽可能多地说些什么。任何大概，本质上的都是一种猜测。任何物理学的理论，都是某种猜测工作。有好的猜测，和坏的猜测。豁然性的理论，是一个系统，旨在做出好的猜测。有些情况，可能有很多变化，但是它们的表现，有一定的一致性和平均性，或然性的语言，允许我们对这些情况，能够定量地说些什么。

Let us consider the flipping of a coin. Ifthe toss—and the coin—are “honest,” we have no way of knowing what to expectfor the outcome of any particular toss. Yet we would feel that in a largenumber of tosses there should be about equal numbers of heads and tails. Wesay: “The probability that a toss will land heads is 0.5 .”
现在让我们考虑抛硬币。如果这个抛掷及硬币，是“诚实的”，那么，对于任何一次具体的抛掷，我们都没有办法知道肯定会是什么结果。尽管如此，我们感觉到，如果抛掷的次数很大，那么，正面和反面的次数，应该相等，我们说：“抛掷一次，得到正面的概率是0.5”。

We speak of probability only forobservations that we contemplate being made in the future. By the“probability” of a particular outcome of an observation we mean our estimatefor the most likely fraction of a number of repeated observations that willyield that particular outcome. If we imagine repeating an observation—such aslooking at a freshly tossed coin—N times, and if we call NA our estimate of the most likely number of our observations thatwill give some specified result A , say the result “heads,” then by P(A) , the probability of observing A , we mean
P(A)=NA/N.(6.1)
只有对那些我们认为将来会发生的观察，我们才会说到概率。对于一次观察的具体结果，我们通过“概率”这个词，所意味的，就是在一次观察中，我们估计能产生某一特别结果的数目的分数。对于一个观察，例如看一个新的抛掷硬币，如果我们想象它重复N次，那么，对于某一种具体结果，例如“正面”的结果，我们把该结果出现的次数，称为NA，所以，通过P(A)，即观察到A出现的概率，我们意味着：
P(A)=NA/N. (6.1)

Our definition requires several comments. First of all, we may speakof a probability of something happening only if the occurrence is a possibleoutcome of some repeatable observation. It is not clear that it wouldmake any sense to ask: “What is the probability that there is a ghost in thathouse?”
我们的定义，需要注释几点。我们观察一件事情，只有当它的某种结果，可能重复出现的时候，才能谈到其概率。反之，问题“在那个房间里有鬼的概率是多少？”有什么意义吗？并不清楚。

You may object that no situation is exactlyrepeatable. That is right. Every different observation must at least be at adifferent time or place. All we can say is that the “repeated” observations should,for our intended purposes, appear to be equivalent. We should assume, atleast, that each observation was made from an equivalently prepared situation,and especially with the same degree of ignorance at the start. (If we sneak alook at an opponent’s hand in a card game, our estimate of our chances ofwinning are different than if we do not!)
We should emphasize that N and NA in Eq. (6.1)are not intended to represent numbers based on actual observations. NAis our best estimate of what would occur in Nimagined observations. Probability depends, therefore, on ourknowledge and on our ability to make estimates. In effect, on our common sense!Fortunately, there is a certain amount of agreement in the common sense of manythings, so that different people will make the same estimate. Probabilitiesneed not, however, be “absolute” numbers. Since they depend on our ignorance,they may become different if our knowledge changes.
你可以反对说：没有任何情况，是完全可以重复的。你说的对。每次不同的观察，至少时间或地点，是不同的。我们所能说的只是：“重复的”观察，应该对于我们的目的来说，显得是一样的。至少，我们应该假定，每次观察出发时的情况，都是一样的，特别是在开始的时候，我们的无知，是同等的。（打牌时，我们能否偷看一下对手的牌，对于我们估计我们是否能赢，要差很多）。我们要强调，方程（6.1）中的N和NA所代表的，并不是基于实际的观察数字。在n次想象的观察中，NA是某结果应该出现的最好的值。因此，概率依赖于我们的知识，及我们估计的能力。实际上，依赖于我们的常识！幸运的是，对于很多事情，大家的常识，相当一致，所以，不同的人，将会做出同样的估计。然而，概率并不需要是“绝对的”数字。鉴于概率依赖于我们的无知，所以，如果我们的知识变了，那么它们也就会不同。

You may have noticed another rather“subjective” aspect of our definition of probability. We have referred to NAas “our estimate of the most likely number …” We do not mean that we expectto observe exactly NA , but that we expect a number near NA , and that the number NA is more likely than any other number in the vicinity. If wetoss a coin, say, 30 times, we should expect that the number of heads would not bevery likely to be exactly 15 , but rather only some number near to 15 , say 12 , 13 , 14 , 15 , 16 , or 17 . However, if we must choose, we would decide that 15 heads is more likely than any other number. We wouldwrite P(heads)=0.5 .
你可能已经注意到了，我们的概率定义中的另一个相当“主观的”方面。我们把NA说成是：我们估计中的最可能的数字。我们的意思，并不是说我们期待观察到准确的NA，但是，我们期待一个接近NA的数，这个数NA，比其附近的数目，更可能些。假设我们掷一个硬币30次。我们期待正面的数目，并不是非常准确的15，而不如说，是一些非常接近15的数目，比如12，13，14，15，16或17。然而，如果我们必须要挑选正面出现的数目的话，我们可能选15，而不是其它的数目。我们会写：P(正面)=0.5。

Why did we choose 15 as more likely than any other number? We must have argued with ourselvesin the following manner: If the most likely number of heads is NH in a total number of tosses N , then the most likely number of tails NT is (N−NH) . (We are assuming that every toss gives either heads ortails, and no “other” result!) But if the coin is “honest,” there is nopreference for heads or tails. Until we have some reason to think the coin (ortoss) is dishonest, we must give equal likelihoods for heads and tails. So wemust set NT=NH . It follows that NT= NH= N/2 , or P(H)= P(T)= 0.5 .
为什么我们更喜欢选择15，而不是其他的数字呢？我们一定以下面的方式，与自己争辩过：在总数为N的抛掷中，最可能的正面数是NH，那么，最可能的反面的数NT，就是N-NH。（我们假定每次抛掷，不是正面，就是反面，没有其他。）但是，如果这个硬币是“诚实的”，那么，对于正面和反面，它就并无偏向。除非我们有理由认为，硬币（或抛掷）是不诚实的，否则我们就应该认为，正面和负面的可能性是同等的。所以，我们就应该设NT=NH。于是可得：NT= NH= N/2 , 或 P(H)=P(T)= 0.5 。

We can generalize our reasoning to anysituation in which there are m different but “equivalent” (that is, equally likely) possible resultsof an observation. If an observation can yield m different results, and we have reason to believe that any one ofthem is as likely as any other, then the probability of a particularoutcome A is P(A)=1/m .
一个观察，如果有m种可能的结果，且这些结果的可能性都是同等的，那么，对于其中的任何一种情况，我们都可以把我们的推理，归纳应用与其上。如果一个观察可以产生m种不同的结果，且我们有理由相信，其中的任何一种，都与其他的一样，那么，某一具体结果A的概率就是： P(A)=1/m。

If there are seven different-colored ballsin an opaque box and we pick one out “at random” (that is, without looking),the probability of getting a ball of a particular color is 1/7. The probability that a “blind draw” from a shuffled deck of 52 cards will show the ten of hearts is 1/52 . The probability of throwing a double-one with dice is 1/36.
如果在一个不透明的盒子里，有七个不同颜色的球，我们“随机地”（即不用看）拿出一个球，那么，拿到某一具体颜色的球的概率为1/7。从一副52张洗过的牌中，“盲目地”抽出一张，它是红心10的概率为1/52。一副骰子，要掷出两个都是一，概率为1/36。

In Chapter 5we described the size of a nucleus in terms of its apparent area, or “crosssection.” When we did so we were really talking about probabilities. When we shoota high-energy particle at a thin slab of material, there is some chance that itwill pass right through and some chance that it will hit a nucleus. (Since thenucleus is so small that we cannot see it, we cannot aim right at anucleus. We must “shoot blind.”) If there are n atoms in our slab and the nucleus of each atom has a cross-sectionalarea σ , then the total area “shadowed” by the nuclei is nσ . In a large number N of random shots, we expect that the number of hits NCof some nucleus will be in the ratio to N as the shadowed area is to the total area of the slab:
NC/N=nσ/A.(6.2)
We may say, therefore, that the probability that any oneprojectile particle will suffer a collision in passing through the slab is
PC=(n/A)σ,(6.3)
where n/A is the number of atoms per unit area in our slab.
在第五章，对于原子核，我们依据其明显的面积、或“横截面积”，讲了其尺寸。我们这么做时，确实谈到了概率。当我们用一个高能粒子，射一个薄的材料版时，它可能通过，也可能撞上一个原子核。（由于原子核是如此之小，以至于我们看不到它，所以，我们不能瞄准一个原子核射，而能“盲射”）。如果我们的薄板中，有n个原子，每个原子的横截面积是σ，那么，被原子核“影住”的总面积就是nσ。在一个数量为N的大数量的随机射击中，我们期待，撞上原子核的粒子数目NC，与N的比，将等于薄板的被影住的面积，与薄板总面积之比：
NC/N=nσ/A.(6.2)
因此，我们可以说，任何一个被抛射的粒子，在通过薄板时，发生碰撞的概率为：
PC=(n/A)σ,(6.3)
这里n/A，是我们板子上单位面积内的原子数。
{？两个公式一样}

6–2Fluctuations 6-2 波动

Fig. 6–1.Observed sequences of heads and tailsin three games of 30 tosses each. 图6-1 三局，每局抛掷硬币30次，所观察到得正面和反面的次序。
We would like now to use our ideas aboutprobability to consider in some greater detail the question: “How many heads doI really expect to get if I toss a coin N times?” Before answering the question, however, let us look atwhat does happen in such an “experiment.” Figure 6–1 showsthe results obtained in the first three “runs” of such an experiment inwhich N=30 . The sequences of “heads” and “tails” are shown just as they wereobtained. The first game gave 11 heads; the second also 11 ; the third 16 . In three trials we did not once get 15 heads. Should we begin to suspect the coin? Or were we wrong in thinkingthat the most likely number of “heads” in such a game is 15 ? Ninety-seven more runs were made to obtain a total of 100 experiments of 30 tosses each. The results of the experiments are given inTable 6–1.1
现在，我们要用我们关于概率的想法，来更深入地考虑这个问题：“如果把一枚硬币抛N次，我期望能得到多少次正面呢？”然而，在我回答这个问题之前，让我们看看，在一个这种实验中，究竟会发生些什么。图6-1显示的，就是一个这种实验的前三次的结果，这里N=30。如图所示，就是所得到的正面和反面的次序。第一局给出11个正面，第二局也是11个，第三局16个。在三局中，我们一次都没有得到过15。我们是否应该怀疑硬币呢？或者，“在这样一种游戏中，最可能的数字是15”这种想法是错的？每次实验抛30次，又做了97次，达到总数100，总的结果如表6-1所示。（脚注1）
Table 6–1Number of heads in successive trials of 30 tosses of a coin. 表6-1 共100次试验，每次试验，一枚硬币抛30次，正面出现的数字。