Xác suất

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Từ xác suất (probability) bắt nguồn từ chữ probare trong tiếng Latinh (nghĩa là để chứng minh, để kiểm chứng). Nói một cách đơn giản, probable là một trong nhiều từ dùng để chỉ những sự kiện hoặc kiến thức chưa chắc chắn, và thường đi kèm với các từ như "có vẻ là", "mạo hiểm", "may rủi", "không chắc chắn" hay "nghi ngờ", tùy vào ngữ cảnh. Cơ hội (chance), cá cược (odds, bet) là những từ cho khái niệm tương tự. Nếu lí thuyết cơ học (cơ học cổ điển) có định nghĩa chính xác cho "công" và "lực", thì lí thuyết xác suất nhằm mục đích định nghĩa "khả năng".

Mục lục

[sửa] Các giai đoạn lịch sử


Khoa học nghiên cứu về xác suất là một ngành khoa học phát triển hiện đại. Việc chơi cờ bạc (gambling) cho chúng ta thấy rằng các ý niệm về xác suất đã có từ trước đây hàng nghìn năm, tuy nhiên các ý niệm đó được mô tả bởi toán học và sử dụng trong thực tế thì có muộn hơn rất nhiều.

Hai nhà toán học nổi tiếng Pierre de Fermat và Blaise Pascal là những người đầu tiên đặt nền móng cho học thuyết về xác suất vào năm (1654). Christiaan Huygens (1657) được biết đến như là người đầu tiên có công trong việc đưa xác suất thành một vấn đề nghiên cứu khoa học. Học thuyết chủ nghĩa về xác suất bắt đầu bằng những lần thư từ qua lại giữa Pierre de Fermat và Blaise Pascal (1654). Christiaan Huygens (1657) đã đưa ra những hiểu biết đầu tiên mang tính khoa học về vấn đề này. Các cuốn sách Ars Conjectandi của Jakob Bernoulli (sau khi chết, 1713) và Học thuyết chủ nghĩa cơ hội (Doctrine of Chances) của Abraham de Moivre (1718) đã xem xét chủ đề này dưới dạng là một nhánh của toán học.

Lý thuyết sai số (the theory of errors) có thể bắt đầu từ cuốn sách Opera Miscellanea của Roger Cotes (sau khi chết, 1722), nhưng trong một luận văn của Thomas Simpson vào năm 1755 (in vào năm 1756) lần đầu tiên áp dụng lí thuyết này vào trong các cuộc thảo luận về lỗi xảy ra khi quan sát. Bản in lại (1757) của luận văn này đưa ra tiên đề rằng khả năng sai số âm và dương (positive and negative errors) là ngang nhau, và rằng có một giới hạn chấp nhận được nào đó mà trong khoảng đó mọi sai số đều xem là không đáng kể; sai số liên tục được thảo luận và một bước ngoặt trong xác suất đã ra đời.

Pierre-Simon Laplace (1774) đã thực hiện nỗ lực đầu tiên trong việc rút ra một qui luật từ việc kết hợp các quan sát từ các nguyên lí của lí thuyết xác suất. Ông đã giới thiệu định luật xác suất về sai số (the law of probability of errors) bằng một đường cong y = φ(x), x là một sai số bất kì và y là xác suất của lỗi đó, và đưa ra 3 thuộc tính cho đường cong này: (1) Nó là đối xứng qua trục y; (2) trục x là đường tiệm cận, xác suất của sai số \infty là 0; (3) diện tích vùng bao phủ là 1, thì một sai số là tồn tại. Ông cũng đã rút ra một công thức từ 3 quan sát đó. Ông cũng đã đưa ra (1781) một công thức cho định luật của điều kiện của sai số (the law of facility of error) (một thuật ngữ của Lagrange, 1774), nhưng công thức này dẫn đến phương trình không thể giải quyết được. Daniel Bernoulli (1778) đã giới thiệu nguyên lí của tích cực đại của các xác suất của một hệ thống sai số đồng thời.

The method of least squares is due to Adrien-Marie Legendre (1805), who introduced it in his Nouvelles méthodes pour la détermination des orbites des comètes (New Methods for Determining the Orbits of Comets). In ignorance of Legendre's contribution, an Irish-American writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,

\phi(x) = ce^{-h^2 x^2}

c and h being constants depending on precision of observation. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof which seems to have been known in Europe (the third after Adrain's) in 1809. Further proofs were given by Laplace (1810, 1812), Gauss (1823), James Ivory (1825, 1826), Hagen (1837), Friedrich Bessel (1838), Donkin (1844, 1856), and Morgan Crofton (1870). Other contributors were Ellis (1844), De Morgan (1864), Glaisher (1872), and Giovanni Schiaparelli (1875). Peters's (1856) formula for r, the probable error of a single observation, is well known.

In the nineteenth century authors on the general theory included Laplace, Sylvestre Lacroix (1816), Littrow (1833), Adolphe Quetelet (1853), Richard Dedekind (1860), Helmert (1872), Hermann Laurent (1873), Liagre, Didion, and Karl Pearson. Augustus De Morgan and George Boole improved the exposition of the theory.

On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).

[sửa] Khái niệm

There is essentially one set of mathematical rules for manipulating probability; these rules are listed under "Formalization of probability" below. (There are other rules for quantifying uncertainty, such as the Dempster-Shafer theory and possibility theory, but those are essentially different and not compatible with the laws of probability as they are usually understood.) However, there is ongoing debate over what, exactly, the rules apply to; this is the topic of probability interpretations.

The general idea of probability is often divided into two related concepts:

  • Aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon. This concept can be further divided into physical phenomena that are predictable, in principle, with sufficient information (see Determinism), and phenomena which are essentially unpredictable. Examples of the first kind include tossing dice or spinning a roulette wheel, and an example of the second kind is radioactive decay.
  • Epistemic probability, which represents our uncertainty about propositions when one lacks complete knowledge of causative circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are to assign a probability to the proposition that a proposed law of physics is true, and to determine how "probable" it is that a suspect committed a crime, based on the evidence presented.

It is an open question whether aleatory probability is reducible to epistemic probability based on our inability to precisely predict every force that might affect the roll of a die, or whether such uncertainties exist in the nature of reality itself, particularly in quantum phenomena governed by Heisenberg's uncertainty principle. Although the same mathematical rules apply regardless of which interpretation is chosen, the choice has major implications for the way in which probability is used to model the real world.

[sửa] Sự hình thành xác suất

Like other theories, the theory of probability is a representation of probabilistic concepts in formal terms -- that is, in terms that can be considered separately from their meaning. These formal terms are manipulated by the rules of mathematics and logic, and any results are then interpreted or translated back into the problem domain.

There have been at least two successful attempts to formalize probability, namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov's formulation, sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's formulation, probability is taken as a primitive (that is, not further analyzed) and the emphasis is on constructing a consistent assignment of probability values to propositions. In both cases, the laws of probability are the same, except for technical details:

  1. a probability is a number between 0 and 1;
  2. the probability of an event or proposition and its complement must add up to 1; and
  3. the joint probability of two events or propositions is the product of the probability of one of them and the probability of the second, conditional on the first.

The reader will find an exposition of the Kolmogorov formulation in the probability theory article, and in the Cox's theorem article for Cox's formulation. See also the article on probability axioms.

For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.

[sửa] Cách biểu diễn và chuyển đổi các giá trị xác suất

The probability of an event is generally represented as a real number between 0 and 1, inclusive. An impossible event has a probability of exactly 0, and a certain event has a probability of 1, but the converses are not always true: probability 0 events are not always impossible, nor probability 1 events certain. The rather subtle distinction between "certain" and "probability 1" is treated at greater length in the article on "almost surely".

Most probabilities that occur in practice are numbers between 0 and 1, indicating the event's position on the continuum between impossibility and certainty. The closer an event's probability is to 1, the more likely it is to occur.

For example, if two mutually exclusive events are assumed equally probable, such as a flipped coin landing heads-up or tails-up, we can express the probability of each event as "1 in 2", or, equivalently, "50%" or "1/2".

Probabilities are equivalently expressed as odds, which is the ratio of the probability of one event to the probability of all other events. The odds of heads-up, for the tossed coin, are (1/2)/(1 - 1/2), which is equal to 1/1. This is expressed as "1 to 1 odds" and often written "1:1".

Odds a:b for some event are equivalent to probability a/(a+b). For example, 1:1 odds are equivalent to probability 1/2, and 3:2 odds are equivalent to probability 3/5.

There remains the question of exactly what can be assigned probability, and how the numbers so assigned can be used; this is the question of probability interpretations. There are some who claim that probability can be assigned to any kind of an uncertain logical proposition; this is the Bayesian interpretation. There are others who argue that probability is properly applied only to random events as outcomes of some specified random experiment, for example sampling from a population; this is the frequentist interpretation. There are several other interpretations which are variations on one or the other of those, or which have less acceptance at present.

[sửa] Sự phân bố

A probability distribution is a function that assigns probabilities to events or propositions. For any set of events or propositions there are many ways to assign probabilities, so the choice of one distribution or another is equivalent to making different assumptions about the events or propositions in question.

There are several equivalent ways to specify a probability distribution. Perhaps the most common is to specify a probability density function. Then the probability of an event or proposition is obtained by integrating the density function. The distribution function may also be specified directly. In one dimension, the distribution function is called the cumulative distribution function. Probability distributions can also be specified via moments or the characteristic function, or in still other ways.

A distribution is called a discrete distribution if it is defined on a countable, discrete set, such as a subset of the integers. A distribution is called a continuous distribution if it has a continuous distribution function, such as a polynomial or exponential function. Most distributions of practical importance are either discrete or continuous, but there are examples of distributions which are neither.

Important discrete distributions include the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution and the Maxwell-Boltzmann distribution.

Important continuous distributions include the normal distribution, the gamma distribution, the Student's t-distribution, and the exponential distribution.

[sửa] Xác suất với toán học

Probability axioms form the basis for mathematical probability theory. Calculation of probabilities can often be determined using combinatorics or by applying the axioms directly. Probability applications include even more than statistics, which is usually based on the idea of probability distributions and the central limit theorem.

To give a mathematical meaning to probability, consider flipping a "fair" coin. Intuitively, the probability that heads will come up on any given coin toss is "obviously" 50%; but this statement alone lacks mathematical rigor - certainly, while we might expect that flipping such a coin 10 times will yield 5 heads and 5 tails, there is no guarantee that this will occur; it is possible for example to flip 10 heads in a row. What then does the number "50%" mean in this context?

One approach is to use the law of large numbers. In this case, we assume that we can perform any number of coin flips, with each coin flip being independent - that is to say, the outcome of each coin flip is unaffected by previous coin flips. If we perform N trials (coin flips), and let NH be the number of times the coin lands heads, then we can, for any N, consider the ratio NH/N.

As N gets larger and larger, we expect that in our example the ratio NH/N will get closer and closer to 1/2. This allows us to "define" the probability Pr(H) of flipping heads as the limit (mathematics), as N approaches infinity, of this sequence of ratios:

\Pr(H) = \lim_{N \to \infty}{N_H \over N}

In actual practice, of course, we cannot flip a coin an infinite number of times; so in general, this formula most accurately applies to situations in which we have already assigned an a priori probability to a particular outcome (in this case, our assumption that the coin was a "fair" coin). The law of large numbers then says that, given Pr(H), and any arbitrarily small number ε, there exists some number n such that for all N > n,

\left| \Pr(H) - {N_H \over N}\right| < \epsilon

In other words, by saying that "the probability of heads is 1/2", we mean that, if we flip our coin often enough, eventually the number of heads over the number of total flips will become arbitrarily close to 1/2; and will then stay at least as close to 1/2 for as long as we keep performing additional coin flips.

Note that a proper definition requires measure theory which provides means to cancel out those cases where the above limit does not provide the "right" result or is even undefined by showing that those cases have a measure of zero.

The a priori aspect of this approach to probability is sometimes troubling when applied to real world situations. For example, in the play Rosencrantz and Guildenstern are Dead by Tom Stoppard, a character flips a coin which keeps coming up heads over and over again, a hundred times. He can't decide whether this is just a random event - after all, it is possible (although unlikely) that a fair coin would give this result - or whether his assumption that the coin is fair is at fault.

[sửa] Những chú ý khi tính toán xác suất

Khó khăn trong việc tính toán xác suất nằm ở việc xác định số sự kiện có thể xảy ra (possible events): đếm số lần xuất hiện của mỗi sự kiện, và đếm số lượng sự kiện có thể xảy ra đó. Đặc biệt khó khăn trong việc rút ra một kết luận có ý nghĩa từ các xác suất tính được. Một bài toán đố thú vị, bài toán Monty Hall sẽ cho thấy điều này.

Để học thêm về cơ bản của lí thuyết xác suất, xem bài viết về tiên đề xác suấtđịnh lý Bayes giải thích việc sử dụng xác suất có điều kiện trong trường hợp sự xuất hiện của 2 sự kiện là có liên quan nhau.

[sửa] Ứng dụng của xác suất với đời sống hàng ngày

A major effect of probability theory on everyday life is in risk assessment and in trade on commodity markets. Governments typically apply probability methods in environment regulation where it is called "pathway analysis", and are often measuring well-being using methods that are stochastic in nature, and choosing projects to undertake based on their perceived probable effect on the population as a whole, statistically. It is not correct to say that statistics are involved in the modelling itself, as typically the assessments of risk are one-time and thus require more fundamental probability models, e.g. "the probability of another 9/11". A law of small numbers tends to apply to all such choices and perception of the effect of such choices, which makes probability measures a political matter.

A good example is the effect of the perceived probability of any widespread Middle East conflict on oil prices - which have ripple effects in the economy as a whole. An assessment by a commodity trade that a war is more likely vs. less likely sends prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are not assessed independently nor necessarily very rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.

It can reasonably be said that the discovery of rigorous methods to assess and combine probability assessments has had a profound effect on modern society. A good example is the application of game theory, itself based strictly on probability, to the Cold War and the mutual assured destruction doctrine. Accordingly, it may be of some importance to most citizens to understand how odds and probability assessments are made, and how they contribute to reputations and to decisions, especially in a democracy.

Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, utilize reliability theory in the design of the product in order to reduce the probability of failure. The probability of failure is also closely associated with the product's warranty.

[sửa] Xem thêm

  • Bayesian probability
  • Bernoulli process
  • Cox's theorem
  • Decision theory
  • Fuzzy measure theory
  • Game of chance
  • Game theory
  • Information theory
  • Law of averages
  • Law of large numbers
  • Measure theory
  • Normal distribution
  • Random fields
  • Random variable
  • Statistics
    • List of statistical topics
  • Stochastic process
  • Wiener process
  • Important publications in probability

[sửa] Liên kết ngoài

[sửa] Các câu nói nổi tiếng

  • Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."
  • Pierre-Simon Laplace "It is remarkable that a science which began with the consideration of games of chance should have become the most important object of human knowledge." Théorie Analytique des Probabilités, 1812.
  • Richard von Mises "The unlimited extension of the validity of the exact sciences was a characteristic feature of the exaggerated rationalism of the eighteenth century" (in reference to Laplace). Probability, Statistics, and Truth, p 9. Dover edition, 1981 (republication of second English edition, 1957).