数学专业英语－Probability

yunjiang · 发表于 2004-5-6 09:48:18

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>The mathematics to which our youngsters are exposed at school is. With rare exceptions, based on the classical yes-or-no, right-or-wrong type of logic. It normally doesn’t include one word about probability as a mode of reasoning or as a basis for comparing several alternative conclusions. Geometry, for instance, is strictly devoted to the “if-then” type of reasoning and so to the notion (idea) that any statement is either correct or incorrect.
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> However, it has been remarked that life is an almost continuous experience of having to draw conclusions from insufficient evidence, and this is what we have to do when we make the trivial decision as to whether or not to carry an umbrella when we leave home for work. This is what a great industry has to do when it decides whether or not to put $50000000 into a new plant abroad. In none of these case and indeed, in practically no other case that you can suggest, can one proceed by saying:” I know that A, B, C, etc. are completely and reliably true, and therefore the inevitable conclusion is~~” For there is another mode of reasoning, which does not say: This statement is correct, and its opposite is completely false.” But which say: There are various alternative possibilities. No one of these is certainly correct and true, and no one certainly incorrect and false. There are varying degrees of plausibility—of probability—for all these alternatives. I can help you understand how these plausibility’s compare; I can also tell you reliable my advice is.”
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> This is the kind of logic, which is developed in the theory of probability. This theory deals with not two truth-values—correct or false—but with all the in intermediate truth values: almost certainly true, very probably true, possibly true, unlikely, very unlikely, etc. Being a precise quantities theory, it does not use phrases such as those just given, but calculates for any question under study the numerical probability that it is true. If the probability has the value of 1, the answer is an unqualified “yes” or certainty. If it is zero (0), the answer is an unqualified “no” i.e. it is false or impossible. If the probability is a half (0.5), then the chances are even that the question has an affirmative answer. If the probability is tenth (0.1), then the chances are only 1 in 10 that the answer is “yes.”
 It is a remarkable fact that one’s intuition is often not very good at csunating answers to probability problems. For ex ample, how many persons must there are at least two persons in the room with the same birthday (born on the same day of the month)? Remembering that there are 356 separate birthdays possible, some persons estimate that there would have to be 50, or even 100, persons in the room to make the odds better than even. The answer, in fact, is that the odds are better than eight to one that at least two will have the same birthday. Let us consider one more example: Everyone is interested in polls, which involve estimating the opinions of a large group (say all those who vote) by determining the opinions of a sample. In statistics the whole group in question is called the “universe” or “population”. Now suppose you want to consult a large enough sample to reflect the whole population with at least 98% precision (accuracy) in 99out of a hundred instances: how large does this very reliable sample have to be? If the population numbers 200 persons, then the sample must include 105 persons, or more than half the whole population. But suppose the population consists of 10,000 persons, or 100,000 persons? In the case of 10,000 persons, or 1000,000 person? In the case of 10,000 persons, a sample, to have the stated reliability, would have to consist of 213 persons: the sample increases by only 108 when the population increases by 9800. And if you add 90000 more to the population, so that it now numbers 100000, you have to add only 4 to the sample. The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of probability rather than on intuition.
 Although the subject started out (began) in the seventeenth century with games of chance such as dice and cards, it soon became clear that it had important applications to other fields of activity. In the eighteenth century Laplace laid the foundations for a theory of errors, and Gauss later develop this into a real working tool for all experimenters and observers. Any measurement or set of measurement is necessarily is necessarily inexact; and it is a matter of the highest importance to know how to take a lot of necessarily discordant data, combine them in the best possible way, and produce in addition some useful estimate of the dependability of the results. Other more modern fields of application are: in life insurance; telephone traffic problems; information and communication theory; game theory, with applications to all forms of competition, including business international politics and war; modern statistical theories, both for the efficient design of experiments and for the interpretation of the results of experiments; decision theories, which aid us in making judgments; probability theories for the process by which we learn, and many more.
 ----Weaver, W.

yunjiang · 发表于 2004-5-6 09:49:01

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0cm 0cm 0pt; TEXT-INDENT: 153pt">Vocabulary<

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robability 概率论 permutation 置换Plausibility 似乎合理 binomial coefficient 二次式系数Affirmative 肯定的 generating function 母函数Estimate 估计 even 事件Discordant 不一致的 information and communication theoryCommunication theory 通讯理论 信息与通讯论Decision theory 决策论 game theory 对策论，博弈论

yunjiang · 发表于 2004-5-6 09:49:16

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0cm 0cm 0pt; TEXT-INDENT: 153pt">Notes<

0cm 0cm 0pt 18pt; TEXT-INDENT: -18pt; tab-stops: list 18.0pt; mso-list: l70 level1 lfo57">1. Geometry, for example, is strictly devoted to the “if—then” type of reasoning and so to the notion (idea) that any statement is either correct or incorrect.<

0cm 0cm 0pt 18pt">意思是：例如几何学就是严格地属于那种“如果，则”的推理类型，所以它也就属于那种对任何陈述要么是对的要么是不对的概念范围。Is devoted to 意思是：“奉献于”，这里可作：“属于”解，注意在and so to the notion~~中，在前面省去is devoted.2. However, it has been remarked that life is an almost continuous experience when we leave home for work.意思是：然而，人们已经注意到，生活就是这样一种几乎不断地需要我们从不充分的证据中去做出结论的经历，这就是对诸如我们离家上班时是否要带雨伞做出定时，我们所需要做的。3. If the probability has value of 1, the answer is an unqualified “yes” or certainty.这里unqualified解作：“绝对的”，“十足的”。如 an unqualified certainty (绝对的肯定)； An unqualified success (彻底胜利)。注意 qualified 常解作：“有资格的”，“合格的”。如 a qualified technician (合格的技术员)； qualified examination (资格考试，美国高等学校研究生院的一种考试)。4． If the probability is a half, then the chances are even that the question has an affirmative answer.意思是：如果概率是一半的话，那么问题有肯定答案的机会是对等的。注意这里even作“对等”解。5． The less credible this seems to you, the more strongly I make the point that it is better to depend on the theory of probability rather than on intuition.意思是：这对你越不可信，我们就要强调这种论点：宁可依靠概率论而不愚信直观，这里make the point that 意思是：“主张；强调；视~~为重要” .

yunjiang · 发表于 2004-5-6 09:49:30

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0cm 0cm 0pt; TEXT-INDENT: 153pt">Exercise<

0cm 0cm 0pt 55.5pt; TEXT-INDENT: -18pt; tab-stops: list 0cm; mso-list: l27 level1 lfo58">1. Translate the following passage into Chinese<

0cm 0cm 0pt 55.5pt">The origin of the theory of probability goes Bach to the mathematical problems connected with dice throwing that were discusses in letters exchanged by B.Ppascal and P.de Fermat in the 17th century. These problems were principally concerned with concepts, such as permutations, combinations, and binomial coefficients, whose theory was established about the same time. This elementary theory of probability was later enriched by the work of scholars such as Jacob Bernoulli, A.de Moivre, T.Bayes, L, de Buffon, Danial Bcrnoulli, A, M, Legendre, and J.L. Lagrange. Finally, P.S. Laplace completed the classical theory of probability in his book “Throrie analytique des probabilities” (1812). In this work, Laplace not only systemized also greatly extended previous important results by introducing new methods such as the use of difference equations and generating functions. Since the 19th century, the theory of probability has been extensively applied to the natural sciences and even to social sciences.2. Translate the following sentences into Chinese:1. The term random process is use to describe process that gives rise to one of a number of admitted possible outcomes but which outcome cannot be predicted with any certainty in advance.2. Tow events A and B in a probability model with sample space and probability function P are said to be independent ifP (A B) =P(A) ·P(B)3. Describe briefly the kind of logic developed in the theory of Probability.4. Translate the following sentences into English (make use of the phrase or the phrases in the bracket):设X=[a b], A X (A X) 是一开集， 又设a A 令r=sup{ : [a+ ] A}, 求证a+r …A. (这一部分不用翻译， 仅需翻译下下面证明部分)证明：（1）若论不成六，即是说a+r A，则由于A是开集，存在 >0使得[a+r, a+r+ ] A, 从而[(a,a+r+ ) A, 这与r 的定义矛盾。（~~~would not hold, or~~~were false, or were not true; contrary to）(2)若a+r A,则由于A是开集，存在 >0使得[a+r,a+r+ ] A，由这推出[a,a+r+ ] A,这是不可能的。故a+r A. (this implies)(3)若论断是错的，则由于A是开集，存在 >0使得[(a+r,a+r+ ) A,从而[a,a+r+ ] A,这就导至与r是 的上确界这一事实相矛盾结论。（leads to contradiction to the that）

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