Probability
Branch of mathematics concerning chance and uncertainty
Probability is a branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. This number is often expressed as a percentage (%), ranging from 0% to 100%. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in "probability theory", which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
Etymology.
The word "probability" derives from the Latin , which can also mean "probity", a measure of the authority of a witness in a legal case in Europe, and often correlated with the witness's nobility. In a sense, this differs much from the modern meaning of "probability", which in contrast is a measure of the weight of empirical evidence, and is arrived at from inductive reasoning and statistical inference.
Interpretations.
When dealing with random experiments – i.e., experiments that are random and well-defined – in a purely theoretical setting (like tossing a coin), probabilities can be numerically described by the number of desired outcomes, divided by the total number of all outcomes. This is referred to as theoretical probability (in contrast to empirical probability, dealing with probabilities in the context of real experiments). For example, tossing a coin twice will yield "head-head", "head-tail", "tail-head", and "tail-tail" outcomes. The probability of getting an outcome of "head-head" is 1 out of 4 outcomes, or, in numerical terms, 1/4, 0.25 or 25%. However, when it comes to practical application, there are two major competing categories of probability interpretations, whose adherents hold different views about the fundamental nature of probability:
History.
The scientific study of probability is a modern development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability throughout history, but exact mathematical descriptions arose much later. There are reasons for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the mathematical study of probability, fundamental issues are still obscured by superstitions.
According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin "probabilis") meant "approvable", and was applied in that sense, univocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances." However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.
The sixteenth-century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes).
Aside from the elementary work by Cardano, the doctrine of probabilities dates to the correspondence of Pierre de Fermat and Blaise Pascal (1654). Christiaan Huygens (1657) gave the earliest known scientific treatment of the subject. Jakob Bernoulli's "Ars Conjectandi" (posthumous, 1713) and Abraham de Moivre's "Doctrine of Chances" (1718) treated the subject as a branch of mathematics. See Ian Hacking's "The Emergence of Probability" and James Franklin's "The Science of Conjecture" for histories of the early development of the very concept of mathematical probability.
The theory of errors may be traced back to Roger Cotes's "Opera Miscellanea" (posthumous, 1722), but a memoir prepared by Thomas Simpson in 1755 (printed 1756) first applied the theory to the discussion of errors of observation. The reprint (1757) of this memoir lays down the axioms that positive and negative errors are equally probable, and that certain assignable limits define the range of all errors. Simpson also discusses continuous errors and describes a probability curve.
The first two laws of error that were proposed both originated with Pierre-Simon Laplace. The first law was published in 1774, and stated that the frequency of an error could be expressed as an exponential function of the numerical magnitude of the error – disregarding sign. The second law of error was proposed in 1778 by Laplace, and stated that the frequency of the error is an exponential function of the square of the error. The second law of error is called the normal distribution or the Gauss law. "It is difficult historically to attribute that law to Gauss, who in spite of his well-known precocity had probably not made this discovery before he was two years old."
Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
Adrien-Marie Legendre (1805) developed the method of least squares, and 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,
formula_1
where formula_2 is a constant depending on precision of observation, and formula_3 is a scale factor ensuring that the area under the curve equals 1. He gave two proofs, the second being essentially the same as John Herschel's (1850). Gauss gave the first proof that 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), W.F. 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.
In 1906, Andrey Markov introduced the notion of Markov chains, which played an important role in stochastic processes theory and its applications. The modern theory of probability based on measure theory was developed by Andrey Kolmogorov in 1931.
On the geometric side, contributors to "The Educational Times" included Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin. See integral geometry for more information.
Theory.
Like other theories, the theory of probability is a representation of its 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 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 (see also probability space), sets are interpreted as events and probability as a measure on a class of sets. In Cox's theorem, probability is taken as a primitive (i.e., 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.
There are other methods for quantifying uncertainty, such as the Dempster–Shafer theory or possibility theory, but those are essentially different and not compatible with the usually-understood laws of probability.
Applications.
Probability theory is applied in everyday life in risk assessment and modeling. The insurance industry and markets use actuarial science to determine pricing and make trading decisions. Governments apply probabilistic methods in environmental regulation, entitlement analysis, and financial regulation.
An example of the use of probability theory in equity trading 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 trader that a war is more likely can send that commodity's prices up or down, and signals other traders of that opinion. Accordingly, the probabilities are neither assessed independently nor necessarily rationally. The theory of behavioral finance emerged to describe the effect of such groupthink on pricing, on policy, and on peace and conflict.
In addition to financial assessment, probability can be used to analyze trends in biology (e.g., disease spread) as well as ecology (e.g., biological Punnett squares). As with finance, risk assessment can be used as a statistical tool to calculate the likelihood of undesirable events occurring, and can assist with implementing protocols to avoid encountering such circumstances. Probability is used to design games of chance so that casinos can make a guaranteed profit, yet provide payouts to players that are frequent enough to encourage continued play.
Another significant application of probability theory in everyday life is reliability. Many consumer products, such as automobiles and consumer electronics, use reliability theory in product design to reduce the probability of failure. Failure probability may influence a manufacturer's decisions on a product's warranty.
The cache language model and other statistical language models that are used in natural language processing are also examples of applications of probability theory.
Mathematical treatment.
Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment, sometimes denoted as formula_4. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a die can produce six possible results. One collection of possible results gives an odd number on the die. Thus, the subset {1,3,5} is an element of the power set of the sample space of dice rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, the event is said to have occurred.
A probability is a way of assigning every event a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) is assigned a value of one. To qualify as a probability, the assignment of values must satisfy the requirement that for any collection of mutually exclusive events (events with no common results, such as the events {1,6}, {3}, and {2,4}), the probability that at least one of the events will occur is given by the sum of the probabilities of all the individual events.
The probability of an event "A" is written as formula_5, formula_6, or formula_7. This mathematical definition of probability can extend to infinite sample spaces, and even uncountable sample spaces, using the concept of a measure.
The "opposite" or "complement" of an event "A" is the event [not "A"] (that is, the event of "A" not occurring), often denoted as formula_8, formula_9, or formula_10; its probability is given by "P"(not "A") = 1 − "P"("A"). As an example, the chance of not rolling a six on a six-sided die is 1 – (chance of rolling a six) = 1 − = . For a more comprehensive treatment, see Complementary event.
If two events "A" and "B" occur on a single performance of an experiment, this is called the intersection or joint probability of "A" and "B", denoted as formula_11
Independent events.
If two events, "A" and "B" are independent then the joint probability is
formula_12
For example, if two coins are flipped, then the chance of both being heads is formula_13
Mutually exclusive events.
If either event "A" or event "B" can occur but never both simultaneously, then they are called mutually exclusive events.
If two events are mutually exclusive, then the probability of "both" occurring is denoted as formula_14 andformula_15If two events are mutually exclusive, then the probability of "either" occurring is denoted as formula_16 andformula_17
For example, the chance of rolling a 1 or 2 on a six-sided die is formula_18
Not (necessarily) mutually exclusive events.
If the events are not (necessarily) mutually exclusive thenformula_19Rewritten,formula_20
For example, when drawing a card from a deck of cards, the chance of getting a heart or a face card (J, Q, K) (or both) is formula_21 since among the 52 cards of a deck, 13 are hearts, 12 are face cards, and 3 are both: here the possibilities included in the "3 that are both" are included in each of the "13 hearts" and the "12 face cards", but should only be counted once.
This can be expanded further for multiple not (necessarily) mutually exclusive events. For three events, this proceeds as follows:formula_22It can be seen, then, that this pattern can be repeated for any number of events.
Conditional probability.
"Conditional probability" is the probability of some event "A", given the occurrence of some other event "B". Conditional probability is written formula_23, and is read "the probability of "A", given "B"". It is defined by
formula_24
If formula_25 then formula_23 is formally undefined by this expression. In this case formula_27 and formula_28 are independent, since formula_29 However, it is possible to define a conditional probability for some zero-probability events, for example by using a σ-algebra of such events (such as those arising from a continuous random variable).
For example, in a bag of 2 red balls and 2 blue balls (4 balls in total), the probability of taking a red ball is formula_30 however, when taking a second ball, the probability of it being either a red ball or a blue ball depends on the ball previously taken. For example, if a red ball was taken, then the probability of picking a red ball again would be formula_31 since only 1 red and 2 blue balls would have been remaining. And if a blue ball was taken previously, the probability of taking a red ball will be formula_32
Inverse probability.
In probability theory and applications, "Bayes' rule" relates the odds of event formula_33 to event formula_34 before (prior to) and after (posterior to) conditioning on another event formula_35 The odds on formula_33 to event formula_37 is simply the ratio of the probabilities of the two events. When arbitrarily many events formula_27 are of interest, not just two, the rule can be rephrased as "posterior is proportional to prior times likelihood", formula_39 where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as formula_27 varies, for fixed or given formula_28 (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005).
Relation to randomness and probability in quantum mechanics.
In a deterministic universe, based on Newtonian concepts, there would be no probability if all conditions were known (Laplace's demon) (but there are situations in which sensitivity to initial conditions exceeds our ability to measure them, i.e. know them). In the case of a roulette wheel, if the force of the hand and the period of that force are known, the number on which the ball will stop would be a certainty (though as a practical matter, this would likely be true only of a roulette wheel that had not been exactly levelled – as Thomas A. Bass' Newtonian Casino revealed). This also assumes knowledge of inertia and friction of the wheel, weight, smoothness, and roundness of the ball, variations in hand speed during the turning, and so forth. A probabilistic description can thus be more useful than Newtonian mechanics for analyzing the pattern of outcomes of repeated rolls of a roulette wheel. Physicists face the same situation in the kinetic theory of gases, where the system, while deterministic "in principle", is so complex (with the number of molecules typically the order of magnitude of the Avogadro constant ) that only a statistical description of its properties is feasible.
Probability theory is required to describe quantum phenomena. A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at sub-atomic scales and are governed by the laws of quantum mechanics. The objective wave function evolves deterministically but, according to the Copenhagen interpretation, it deals with probabilities of observing, the outcome being explained by a wave function collapse when an observation is made. However, the loss of determinism for the sake of instrumentalism did not meet with universal approval. Albert Einstein famously in a letter to Max Born: "I am convinced that God does not play dice". Like Einstein, Erwin Schrödinger, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality. In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes.
Notes.
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References.
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Probability (disambiguation)
Probability is the measure of an event's likelihood.
Probability may also refer to:
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Luck (disambiguation)
Luck is a chance happening.
Luck may also refer to:
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Chance
Chance may refer to:
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Outline of probability
Overview of and topical guide to probability
Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition whose truth is not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain is it that the event will occur?" The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
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Chance of Rain
Chance of Rain may refer to:
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Doctrine of chances (disambiguation)
The term doctrine of chances is any of several things:
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Conditional probability
Probability of an event occurring, given that another event has already occurred
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This particular method relies on event A occurring with some sort of relationship with another event B. In this situation, the event A can be analyzed by a conditional probability with respect to B. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P("A"|"B") or occasionally P"B"("A"). This can also be understood as the fraction of probability B that intersects with A, or the ratio of the probabilities of both events happening to the "given" one happening (how many times A occurs rather than not assuming B has occurred): formula_1.
For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone sick is coughing might be 75%, in which case we would have that P(Cough) = 5% and P(Cough|Sick) = 75 %. Although there is a relationship between A and B in this example, such a relationship or dependence between A and B is not necessary, nor do they have to occur simultaneously.
P("A"|"B") may or may not be equal to P("A"), i.e., the unconditional probability or absolute probability of A. If P("A"|"B") = P("A"), then events A and B are said to be "independent": in such a case, knowledge about either event does not alter the likelihood of each other. P("A"|"B") (the conditional probability of A given B) typically differs from P("B"|"A"). For example, if a person has dengue fever, the person might have a 90% chance of being tested as positive for the disease. In this case, what is being measured is that if event B ("having dengue") has occurred, the probability of A ("tested as positive") given that B occurred is 90%, simply writing P("A"|"B") = 90%. Alternatively, if a person is tested as positive for dengue fever, they may have only a 15% chance of actually having this rare disease due to high false positive rates. In this case, the probability of the event B ("having dengue") given that the event A ("testing positive") has occurred is 15% or P("B"|"A") = 15%. It should be apparent now that falsely equating the two probabilities can lead to various errors of reasoning, which is commonly seen through base rate fallacies.
While conditional probabilities can provide extremely useful information, limited information is often supplied or at hand. Therefore, it can be useful to reverse or convert a conditional probability using Bayes' theorem: formula_2. Another option is to display conditional probabilities in a conditional probability table to illuminate the relationship between events.
Definition.
Conditioning on an event.
Kolmogorov definition.
Given two events A and B from the sigma-field of a probability space, with the unconditional probability of B being greater than zero (i.e., P("B") > 0), the conditional probability of A given B (formula_3) is the probability of "A" occurring if "B" has or is assumed to have happened. "A" is assumed to be the set of all possible outcomes of an experiment or random trial that has a restricted or reduced sample space. The conditional probability can be found by the quotient of the probability of the joint intersection of events A and B, that is, formula_4, the probability at which "A" and "B" occur together, and the probability of B:
formula_1.
For a sample space consisting of equal likelihood outcomes, the probability of the event "A" is understood as the fraction of the number of outcomes in "A" to the number of all outcomes in the sample space. Then, this equation is understood as the fraction of the set formula_6 to the set "B". Note that the above equation is a definition, not just a theoretical result. We denote the quantity formula_7 as formula_8 and call it the "conditional probability of A given B."
As an axiom of probability.
Some authors, such as de Finetti, prefer to introduce conditional probability as an axiom of probability:
formula_9.
This equation for a conditional probability, although mathematically equivalent, may be intuitively easier to understand. It can be interpreted as "the probability of "B" occurring multiplied by the probability of "A" occurring, provided that "B" has occurred, is equal to the probability of the "A" and "B" occurrences together, although not necessarily occurring at the same time". Additionally, this may be preferred philosophically; under major probability interpretations, such as the subjective theory, conditional probability is considered a primitive entity. Moreover, this "multiplication rule" can be practically useful in computing the probability of formula_6 and introduces a symmetry with the summation axiom for Poincaré Formula:
formula_11
Thus the equations can be combined to find a new representation of the :
formula_12
formula_13
As the probability of a conditional event.
Conditional probability can be defined as the probability of a conditional event formula_14. The Goodman–Nguyen–Van Fraassen conditional event can be defined as:
formula_15 and formula_16 represent states or elements of "A" or "B."
It can be shown that
formula_17
which meets the Kolmogorov definition of conditional probability.
Conditioning on an event of probability zero.
If formula_18, then according to the definition, formula_19 is undefined.
The case of greatest interest is that of a random variable Y, conditioned on a continuous random variable X resulting in a particular outcome x. The event formula_20 has probability zero and, as such, cannot be conditioned on.
Instead of conditioning on X being "exactly" x, we could condition on it being closer than distance formula_21 away from x. The event formula_22 will generally have nonzero probability and hence, can be conditioned on.
We can then take the limit
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For example, if two continuous random variables X and Y have a joint density formula_23, then by L'Hôpital's rule and Leibniz integral rule, upon differentiation with respect to formula_21:
formula_25
The resulting limit is the conditional probability distribution of Y given X and exists when the denominator, the probability density formula_26, is strictly positive.
It is tempting to "define" the undefined probability formula_27 using limit (1), but this cannot be done in a consistent manner. In particular, it is possible to find random variables X and W and values x, w such that the events formula_28 and formula_29 are identical but the resulting limits are not:
formula_30
The Borel–Kolmogorov paradox demonstrates this with a geometrical argument.
Conditioning on a discrete random variable.
Let X be a discrete random variable and its possible outcomes denoted V. For example, if X represents the value of a rolled dice then V is the set formula_31. Let us assume for the sake of presentation that X is a discrete random variable, so that each value in V has a nonzero probability.
For a value x in V and an event A, the conditional probability
is given by formula_32.
Writing
formula_33
for short, we see that it is a function of two variables, x and A.
For a fixed A, we can form the random variable formula_34. It represents an outcome of formula_32 whenever a value x of X is observed.
The conditional probability of A given X can thus be treated as a random variable Y with outcomes in the interval formula_36. From the law of total probability, its expected value is equal to the unconditional probability of A.
Partial conditional probability.
The partial conditional probability
formula_37
is about the probability of event
formula_38 given that each of the condition events
formula_39 has occurred to a degree
formula_40 (degree of belief, degree of experience) that might be different from 100%. Frequentistically, partial conditional probability makes sense, if the conditions are tested in experiment repetitions of appropriate length
formula_41. Such
formula_41-bounded partial conditional probability can be defined as the conditionally expected average occurrence of event
formula_38 in testbeds of length
formula_41 that adhere to all of the probability specifications
formula_45, i.e.:
formula_46
Based on that, partial conditional probability can be defined as
formula_47
where formula_48
Jeffrey conditionalization
is a special case of partial conditional probability, in which the condition events must form a partition:
formula_49
Example.
Suppose that somebody secretly rolls two fair six-sided dice, and we wish to compute the probability that the face-up value of the first one is 2, given the information that their sum is no greater than 5.
"Probability that" "D"1 = 2
Table 1 shows the sample space of 36 combinations of rolled values of the two dice, each of which occurs with probability 1/36, with the numbers displayed in the red and dark gray cells being "D"1 + "D"2.
"D"1 = 2 in exactly 6 of the 36 outcomes; thus "P"("D"1 = 2) = <templatestyles src="Fraction/styles.css" />6⁄36 = <templatestyles src="Fraction/styles.css" />1⁄6:
"Probability that" "D"1 + "D"2 ≤ 5
Table 2 shows that "D"1 + "D"2 ≤ 5 for exactly 10 of the 36 outcomes, thus "P"("D"1 + "D"2 ≤ 5) = <templatestyles src="Fraction/styles.css" />10⁄36:
"Probability that" "D"1 = 2 "given that" "D"1 + "D"2 ≤ 5
Table 3 shows that for 3 of these 10 outcomes, "D"1 = 2.
Thus, the conditional probability P("D"1 = 2 | "D"1+"D"2 ≤ 5) = <templatestyles src="Fraction/styles.css" />3⁄10 = 0.3:
Here, in the earlier notation for the definition of conditional probability, the conditioning event "B" is that "D"1 + "D"2 ≤ 5, and the event "A" is "D"1 = 2. We have formula_50 as seen in the table.
Use in inference.
In statistical inference, the conditional probability is an update of the probability of an event based on new information. The new information can be incorporated as follows:
This approach results in a probability measure that is consistent with the original probability measure and satisfies all the Kolmogorov axioms. This conditional probability measure also could have resulted by assuming that the relative magnitude of the probability of "A" with respect to "X" will be preserved with respect to "B" (cf. a Formal Derivation below).
The wording "evidence" or "information" is generally used in the Bayesian interpretation of probability. The conditioning event is interpreted as evidence for the conditioned event. That is, "P"("A") is the probability of "A" before accounting for evidence "E", and "P"("A"|"E") is the probability of "A" after having accounted for evidence "E" or after having updated "P"("A"). This is consistent with the frequentist interpretation, which is the first definition given above.
Example.
When Morse code is transmitted, there is a certain probability that the "dot" or "dash" that was received is erroneous. This is often taken as interference in the transmission of a message. Therefore, it is important to consider when sending a "dot", for example, the probability that a "dot" was received. This is represented by: formula_54 In Morse code, the ratio of dots to dashes is 3:4 at the point of sending, so the probability of a "dot" and "dash" are formula_55. If it is assumed that the probability that a dot is transmitted as a dash is 1/10, and that the probability that a dash is transmitted as a dot is likewise 1/10, then Bayes's rule can be used to calculate formula_56.
formula_57
formula_58
formula_59
Now, formula_60 can be calculated:
formula_61
Statistical independence.
Events "A" and "B" are defined to be statistically independent if the probability of the intersection of A and B is equal to the product of the probabilities of A and B:
formula_62
If "P"("B") is not zero, then this is equivalent to the statement that
formula_63
Similarly, if "P"("A") is not zero, then
formula_64
is also equivalent. Although the derived forms may seem more intuitive, they are not the preferred definition as the conditional probabilities may be undefined, and the preferred definition is symmetrical in "A" and "B". Independence does not refer to a disjoint event.
It should also be noted that given the independent event pair [A B] and an event C, the pair is defined to be conditionally independent if the product holds true:
formula_65
This theorem could be useful in applications where multiple independent events are being observed.
Independent events vs. mutually exclusive events
The concepts of mutually independent events and mutually exclusive events are separate and distinct. The following table contrasts results for the two cases (provided that the probability of the conditioning event is not zero).
In fact, mutually exclusive events cannot be statistically independent (unless both of them are impossible), since knowing that one occurs gives information about the other (in particular, that the latter will certainly not occur).
"These fallacies should not be confused with Robert K. Shope's 1978 "conditional fallacy", which deals with counterfactual examples that beg the question."
Common fallacies.
Assuming conditional probability is of similar size to its inverse.
In general, it cannot be assumed that "P"("A"|"B") ≈ "P"("B"|"A"). This can be an insidious error, even for those who are highly conversant with statistics. The relationship between "P"("A"|"B") and "P"("B"|"A") is given by Bayes' theorem:
formula_66
That is, "P"("A"|"B") ≈ "P"("B"|"A") only if "P"("B")/"P"("A") ≈ 1, or equivalently, "P"("A") ≈ "P"("B").
Assuming marginal and conditional probabilities are of similar size.
In general, it cannot be assumed that "P"("A") ≈ "P"("A"|"B"). These probabilities are linked through the law of total probability:
formula_67
where the events formula_68 form a countable partition of formula_69.
This fallacy may arise through selection bias. For example, in the context of a medical claim, let "S""C" be the event that a sequela (chronic disease) "S" occurs as a consequence of circumstance (acute condition) "C". Let "H" be the event that an individual seeks medical help. Suppose that in most cases, "C" does not cause "S" (so that "P"("S""C") is low). Suppose also that medical attention is only sought if "S" has occurred due to "C". From experience of patients, a doctor may therefore erroneously conclude that "P"("S""C") is high. The actual probability observed by the doctor is "P"("S""C"|"H").
Over- or under-weighting priors.
Not taking prior probability into account partially or completely is called "base rate neglect". The reverse, insufficient adjustment from the prior probability is "conservatism".
Formal derivation.
Formally, "P"("A" | "B") is defined as the probability of "A" according to a new probability function on the sample space, such that outcomes not in "B" have probability 0 and that it is consistent with all original probability measures.
Let Ω be a discrete sample space with elementary events {"ω"}, and let "P" be the probability measure with respect to the σ-algebra of Ω. Suppose we are told that the event "B" ⊆ Ω has occurred. A new probability distribution (denoted by the conditional notation) is to be assigned on {"ω"} to reflect this. All events that are not in "B" will have null probability in the new distribution. For events in "B", two conditions must be met: the probability of "B" is one and the relative magnitudes of the probabilities must be preserved. The former is required by the axioms of probability, and the latter stems from the fact that the new probability measure has to be the analog of "P" in which the probability of "B" is one - and every event that is not in "B", therefore, has a null probability. Hence, for some scale factor "α", the new distribution must satisfy:
Substituting 1 and 2 into 3 to select "α":
formula_73
So the new probability distribution is
Now for a general event "A",
formula_76
See also.
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References.
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A Sporting Chance
A Sporting Chance or Sporting Chance may refer to:
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Event (probability theory)
In statistics and probability theory, set of outcomes to which a probability is assigned
In probability theory, an event is a subset of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an elementary event or an atomic event; that is, it is a singleton set. An event that has more than one possible outcome is called a compound event. An event formula_1 is said to occur if formula_1 contains the outcome formula_3 of the experiment (or trial) (that is, if formula_4). The probability (with respect to some probability measure) that an event formula_1 occurs is the probability that formula_1 contains the outcome formula_3 of an experiment (that is, it is the probability that formula_4).
An event defines a complementary event, namely the complementary set (the event not occurring), and together these define a Bernoulli trial: did the event occur or not?
Typically, when the sample space is finite, any subset of the sample space is an event (that is, all elements of the power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see , below).
A simple example.
If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:
Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically using Venn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability formula_9 of an event formula_10 is the following <templatestyles src="Template:Visible anchor/styles.css" />formula:
formula_11
This rule can readily be applied to each of the example events above.
Events in probability spaces.
Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution, the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers 'badly behaved' sets, such as those that are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice of σ-algebra is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice.
In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected 𝜎-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest are elements of the 𝜎-algebra.
A note on notation.
Even though events are subsets of some sample space formula_12 they are often written as predicates or indicators involving random variables. For example, if formula_13 is a real-valued random variable defined on the sample space formula_12 the event
formula_15
can be written more conveniently as, simply,
formula_16
This is especially common in formulas for a probability, such as
formula_17
The set formula_18 is an example of an inverse image under the mapping formula_13 because formula_20 if and only if formula_21
Notes.
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Expected value (disambiguation)
Expected value is a term used in probability theory and statistics. It may also refer to:
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Chance Meeting
Chance Meeting may refer to:
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Odds
Ratio of the probability of an event happening versus not happening
In probability theory, odds provide a measure of the probability of a particular outcome. Odds are commonly used in gambling and statistics. For example for an event that is 40% probable, one could say that the odds are "2 in 5", "2 to 3 in favor", "2 to 3 on", or "3 to 2 against".
When gambling, odds are often given as the ratio of the possible net profit "to" the possible net loss. However in many situations, you pay the possible loss ("stake" or "wager") up front and, if you win, you are paid the net win plus you also get your stake returned. So wagering 2 at "3 "to" 2", pays out 3 + 2 = 5, which is called "5 "for" 2". When Moneyline odds are quoted as a positive number +"X", it means that a wager pays When Moneyline odds are quoted as a negative number −"X", it means that a wager pays
Odds have a simple relationship with probability. When probability is expressed as a number between 0 and 1, the relationships between probability p and odds are as follows. Note that if probability is to be expressed as a percentage these probability values should be multiplied by 100%.
The numbers for odds can be scaled. If k is any positive number then is the same as and similarly if "to" is replaced with "in" or "for". For example, "3 to 2 against" is the same as both "1.5 to 1 against" and "6 to 4 against".
When the value of the probability p (between 0 and 1; not a percentage) can be written as a fraction "N" / "D" then the odds can be said to be or and these can be scaled to equivalent odds. Similarly, fair betting odds can be expressed as or
History.
The language of odds, such as the use of phrases like "ten to one" for intuitively estimated risks, is found in the sixteenth century, well before the development of probability theory. Shakespeare wrote:
<templatestyles src="Template:Blockquote/styles.css" />Knew that we ventured on such dangerous seas
That if we wrought out life 'twas ten to one
The sixteenth-century polymath Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes. Implied by this definition is the fact that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes.
Statistical usage.
In statistics, odds are an expression of relative probabilities, generally quoted as the odds "in favor". The odds (in favor) of an event or a proposition is the ratio of the probability that the event will happen to the probability that the event will not happen. Mathematically, this is a Bernoulli trial, as it has exactly two outcomes. In case of a finite sample space of equally probable outcomes, this is the ratio of the number of outcomes where the event occurs to the number of outcomes where the event does not occur; these can be represented as "W" and "L" (for Wins and Losses) or "S" and "F" (for Success and Failure). For example, the odds that a randomly chosen day of the week is during a weekend are two to five (2:5), as days of the week form a sample space of seven outcomes, and the event occurs for two of the outcomes (Saturday and Sunday), and not for the other five. Conversely, given odds as a ratio of integers, this can be represented by a probability space of a finite number of equally probable outcomes. These definitions are equivalent, since dividing both terms in the ratio by the number of outcomes yields the probabilities: formula_1 Conversely, the odds against is the opposite ratio. For example, the odds against a random day of the week being during a weekend are 5:2.
Odds and probability can be expressed in prose via the prepositions "to" and "in:" "odds of so many "to" so many on (or against) [some event]" refers to "odds"—the ratio of numbers of (equally probable) outcomes in favor and against (or vice versa); "chances of so many [outcomes], "in" so many [outcomes]" refers to "probability"—the number of (equally probable) outcomes in favour relative to the number for and against combined. For example, "odds of a weekend are 2 "to" 5", while "chances of a weekend are 2 "in" 7". In casual use, the words "odds" and "chances" (or "chance") are often used interchangeably to vaguely indicate some measure of odds or probability, though the intended meaning can be deduced by noting whether the preposition between the two numbers is "to" or "in".
Mathematical relations.
Odds can be expressed as a ratio of two numbers, in which case it is not unique—scaling both terms by the same factor does not change the proportions: 1:1 odds and 100:100 odds are the same (even odds). Odds can also be expressed as a number, by dividing the terms in the ratio—in this case it is unique (different fractions can represent the same rational number). Odds as a ratio, odds as a number, and probability (also a number) are related by simple formulas, and similarly odds in favor and odds against, and probability of success and probability of failure have simple relations. Odds range from 0 to infinity, while probabilities range from 0 to 1, and hence are often represented as a percentage between 0% and 100%: reversing the ratio switches odds for with odds against, and similarly probability of success with probability of failure.
Given odds (in favor) as the ratio W:L (number of outcomes that are wins:number of outcomes that are losses), the odds in favor (as a number) formula_2 and odds against (as a number) formula_3 can be computed by simply dividing, and are multiplicative inverses:
formula_4
Analogously, given odds as a ratio, the probability of success p or failure q can be computed by dividing, and the probability of success and probability of failure sum to unity (one), as they are the only possible outcomes. In case of a finite number of equally probable outcomes, this can be interpreted as the number of outcomes where the event occurs divided by the total number of events:
formula_5
Given a probability "p," the odds as a ratio is formula_6 (probability of success to probability of failure), and the odds as numbers can be computed by dividing:
formula_7
Conversely, given the odds as a number formula_8 this can be represented as the ratio formula_9 or conversely formula_10 from which the probability of success or failure can be computed:
formula_11
Thus if expressed as a fraction with a numerator of 1, probability and odds differ by exactly 1 in the denominator: a probability of 1 "in" 100 (1/100 = 1%) is the same as odds of 1 "to" 99 (1/99 = 0.0101... = 0.01), while odds of 1 "to" 100 (1/100 = 0.01) is the same as a probability of 1 "in" 101 (1/101 = 0.00990099... = 0.0099). This is a minor difference if the probability is small (close to zero, or "long odds"), but is a major difference if the probability is large (close to one).
These are worked out for some simple odds:
These transforms have certain special geometric properties: the conversions between odds for and odds against (resp. probability of success with probability of failure) and between odds and probability are all Möbius transformations (fractional linear transformations). They are thus specified by three points (sharply 3-transitive). Swapping odds for and odds against swaps 0 and infinity, fixing 1, while swapping probability of success with probability of failure swaps 0 and 1, fixing .5; these are both order 2, hence circular transforms. Converting odds to probability fixes 0, sends infinity to 1, and sends 1 to .5 (even odds are 50% probable), and conversely; this is a parabolic transform.
Applications.
In probability theory and statistics, odds and similar ratios may be more natural or more convenient than probabilities. In some cases the log-odds are used, which is the logit of the probability. Most simply, odds are frequently multiplied or divided, and log converts multiplication to addition and division to subtractions. This is particularly important in the logistic model, in which the log-odds of the target variable are a linear combination of the observed variables.
Similar ratios are used elsewhere in statistics; of central importance is the likelihood ratio in likelihoodist statistics, which is used in Bayesian statistics as the Bayes factor.
Odds are particularly useful in problems of sequential decision making, as for instance in problems of how to stop (online) on a last specific event which is solved by the odds algorithm.
The odds are a ratio of probabilities; an odds ratio is a ratio of odds, that is, a ratio of ratios of probabilities. Odds-ratios are often used in analysis of clinical trials. While they have useful mathematical properties, they can produce counter-intuitive results: an event with an 80% probability of occurring is four times "more probable" to happen than an event with a 20% probability, but the "odds" are 16 times higher on the less probable event (4–1 "against", or 4) than on the more probable one (1–4 "against", 4-1 "in favor", 4–1 "on", or 0.25).
Answer: The odds in favour of a blue marble are 2:13. One can equivalently say that the odds are 13:2 "against". There are 2 out of 15 chances in favour of blue, 13 out of 15 against blue.
In probability theory and statistics, where the variable "p" is the probability in favor of a binary event, and the probability against the event is therefore 1-"p", "the odds" of the event are the quotient of the two, or formula_12. That value may be regarded as the relative probability the event will happen, expressed as a fraction (if it is less than 1), or a multiple (if it is equal to or greater than one) of the likelihood that the event will not happen.
In the first example at top, saying the odds of a Sunday are "one to six" or, less commonly, "one-sixth" means the probability of picking a Sunday randomly is one-sixth the probability of not picking a Sunday. While the mathematical probability of an event has a value in the range from zero to one, "the odds" in favor of that same event lie between zero and infinity. The odds against the event with probability given as "p" are formula_13. The odds against Sunday are 6:1 or 6/1 = 6. It is 6 times as probable that a random day is not a Sunday.
Gambling usage.
On a coin toss or a match race between two evenly matched horses, it is reasonable for two people to wager level stakes. However, in more variable situations, such as a multi-runner horse race or a football match between two unequally matched teams, betting "at odds" provides the possibility to take the respective likelihoods of the possible outcomes into account. The use of odds in gambling facilitates betting on events where the probabilities of different outcomes vary.
In the modern era, most fixed-odd betting takes place between a betting organisation, such as a bookmaker, and an individual, rather than between individuals. Different traditions have grown up in how to express odds to customers.
Fractional odds.
Favoured by bookmakers in the United Kingdom and Ireland, and also common in horse racing, fractional odds quote the net total that will be paid out to the bettor, should they win, relative to the stake. Odds of 4/1 ("4 to 1 against") would imply that the bettor stands to make a £400 profit on a £100 stake. If the odds are 1/4 ("1 to 4 against", "4 to 1 in favor", or "4 to 1 on"), the bettor will make £25 on a £100 stake. In either case, having won, the bettor always receives the original stake back; so if the odds are 4/1 the bettor receives a total of £500 (£400 plus the original £100). Odds of 1/1 are known as "evens" or "even money".
The numerator and denominator of fractional odds are often integers, thus if the bookmaker's payout was to be £1.25 for every £1 stake, this would be equivalent to £5 for every £4 staked, and the odds would therefore be expressed as 5/4. However, not all fractional odds are traditionally read using the lowest common denominator. For example, given that there is a pattern of odds of 5/4, 7/4, 9/4 and so on, odds which are mathematically 3/2 are more easily compared if expressed in the equivalent form 6/4.
Fractional odds are also known as "British odds," "UK odds," or, in that country, "traditional odds". They are typically represented with a "/" but can also be represented with a "-", e.g. 4/1 or 4–1. Odds with a denominator of 1 are often presented in listings as the numerator only.
A variation of fractional odds is known as "Hong Kong" odds. Fractional and Hong Kong odds are actually exchangeable. The only difference is that the UK odds are presented as a fractional notation (e.g. 6/5) whilst the Hong Kong odds are decimal (e.g. 1.2). Both exhibit the net return.
Decimal odds.
The European odds also represent the potential winnings (net returns), but in addition they factor in the stake (e.g. 6/5 or 1.2 plus 1 = 2.2).
Favoured in continental Europe, Australia, New Zealand, Canada, and Singapore, decimal odds quote the ratio of the payout amount, "including" the original stake, to the stake itself. Therefore, the decimal odds of an outcome are equivalent to the decimal value of the fractional odds plus one. Thus even odds 1/1 are quoted in decimal odds as 2.00. The 4/1 fractional odds discussed above are quoted as 5.00, while the 1/4 odds are quoted as 1.25. This is considered to be ideal for parlay betting, because the odds to be paid out are simply the product of the odds for each outcome wagered on. When looking at decimal odds in betting terms, the underdog has the higher of the two decimals, while the favorite has the lower of the two. To calculate decimal odds, you can use the equation "Payout = Initial Wager × Decimal Value""." For example, if you bet €100 on Liverpool to beat Manchester City at 2.00 odds the payout, including your stake, would be €200 (€100 × 2.00). Decimal odds are favoured by betting exchanges because they are the easiest to work with for trading, as they reflect the reciprocal of the probability of an outcome. For example, a quoted odds of 5.00 equals to a probability of 1 / 5.00, that is 0.20 or 20%.
Decimal odds are also known as "European odds", "digital odds" or "continental odds."
Moneyline odds.
Moneyline odds are favoured by American bookmakers. The figure quoted is either positive or negative.
Moneyline odds are often referred to as "American odds". A "moneyline" wager refers to odds on the straight-up outcome of a game with no consideration to a point spread. In most cases, the favorite will have negative moneyline odds (less payoff for a safer bet) and the underdog will have positive moneyline odds (more payoff for a risky bet). However, if the teams are evenly matched, "both" teams can have a negative line at the same time (e.g. −110 −110 or −105 −115), due to house take.
Wholesale odds.
Wholesale odds are the "real odds" or 100% probability of an event occurring. This 100% book is displayed without any bookmaker's profit margin, often referred to as a bookmaker's "overround" built in.
A "wholesale odds" index is an index of all the prices in a probabilistic market operating at 100% competitiveness and displayed without any profit margin factored for market participants.
Gambling odds vis-à-vis probabilities.
In gambling, the odds on display do not represent the true chances (as imagined by the bookmaker) that the event will or will not occur, but are the amount that the bookmaker will pay out on a winning bet, together with the required stake. In formulating the odds to display the bookmaker will have included a profit margin which effectively means that the payout to a successful bettor is less than that represented by the true chance of the event occurring. This profit is known as the 'overround' on the 'book' (the 'book' refers to the old-fashioned ledger in which wagers were recorded, and is the derivation of the term 'bookmaker') and relates to the sum of the 'odds' in the following way:
In a 3-horse race, for example, the true probabilities of each of the horses winning based on their relative abilities may be 50%, 40% and 10%. The total of these three percentages is 100%, thus representing a fair 'book'. The true odds against winning for each of the three horses are 1–1, 3–2 and 9–1, respectively.
In order to generate a profit on the wagers accepted, the bookmaker may decide to increase the values to 60%, 50% and 20% for the three horses, respectively. This represents the odds against each, which are 4–6, 1–1 and 4–1, in order. These values now total 130%, meaning that the book has an overround of 30 (130−100). This value of 30 represents the amount of profit for the bookmaker if he gets bets in good proportions on each of the horses. For example, if he takes £60, £50, and £20 of stakes, respectively, for the three horses, he receives £130 in wagers but only pays £100 back (including stakes), whichever horse wins. And the expected value of his profit is positive even if everybody bets on the same horse. The art of bookmaking is in setting the odds low enough so as to have a positive expected value of profit while keeping the odds high enough to attract customers, and at the same time attracting enough bets for each outcome to reduce his risk exposure.
A study on soccer betting found that the probability for the home team to win was generally about 3.4% less than the value calculated from the odds (for example, 46.6% for even odds). It was about 3.7% less for wins by the visitors, and 5.7% less for draws.
To understand roulette probabilities and calculate them, you need to know the formula. You take the numbers your bet is on and divide them by the total number of numbers in roulette (depending on your version of the game). Then you multiply by 100.
Making a profit in gambling involves predicting the relationship of the true probabilities to the payout odds. Sports information services are often used by professional and semi-professional sports bettors to help achieve this goal.
The odds or amounts the bookmaker will pay are determined by the total amount that has been bet on all of the possible events. They reflect the balance of wagers on either side of the event, and include the deduction of a bookmaker's brokerage fee ("vig" or vigorish).
Also, depending on how the betting is affected by jurisdiction, taxes may be involved for the bookmaker and/or the winning player. This may be taken into account when offering the odds and/or may reduce the amount won by a player.
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意外
意外 meaning "accident".
It may refer to:
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Happenstance
Happenstance, a portmanteau of "happening" and "circumstance", may refer to:
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Almost surely
Probability saying
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between "almost surely" and "surely" (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0.
Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of "almost surely": an event that happens with probability zero happens "almost never".
Formal definition.
Let formula_1 be a probability space. An event formula_2 happens "almost surely" if formula_3. Equivalently, formula_4 happens almost surely if the probability of formula_4 not occurring is zero: formula_6. More generally, any set formula_7 (not necessarily in formula_8) happens almost surely if formula_9 is contained in a null set: a subset formula_10 in formula_11 such that The notion of almost sureness depends on the probability measure formula_12. If it is necessary to emphasize this dependence, it is customary to say that the event formula_4 occurs "P"-almost surely, or almost surely "formula_14".
Illustrative examples.
In general, an event can happen "almost surely", even if the probability space in question includes outcomes which do not belong to the event—as the following examples illustrate.
Throwing a dart.
Imagine throwing a dart at a unit square (a square with an area of 1) so that the dart always hits an exact point in the square, in such a way that each point in the square is equally likely to be hit. Since the square has area 1, the probability that the dart will hit any particular subregion of the square is equal to the area of that subregion. For example, the probability that the dart will hit the right half of the square is 0.5, since the right half has area 0.5.
Next, consider the event that the dart hits exactly a point in the diagonals of the unit square. Since the area of the diagonals of the square is 0, the probability that the dart will land exactly on a diagonal is 0. That is, the dart will "almost never" land on a diagonal (equivalently, it will "almost surely" not land on a diagonal), even though the set of points on the diagonals is not empty, and a point on a diagonal is no less possible than any other point.
Tossing a coin repeatedly.
Consider the case where a (possibly biased) coin is tossed, corresponding to the probability space formula_15, where the event formula_16 occurs if a head is flipped, and formula_17 if a tail is flipped. For this particular coin, it is assumed that the probability of flipping a head is formula_18, from which it follows that the complement event, that of flipping a tail, has probability formula_19.
Now, suppose an experiment were conducted where the coin is tossed repeatedly, with outcomes formula_20 and the assumption that each flip's outcome is independent of all the others (i.e., they are independent and identically distributed; "i.i.d"). Define the sequence of random variables on the coin toss space, formula_21 where formula_22. "i.e." each formula_23 records the outcome of the formula_24th flip.
In this case, any infinite sequence of heads and tails is a possible outcome of the experiment. However, any particular infinite sequence of heads and tails has probability 0 of being the exact outcome of the (infinite) experiment. This is because the "i.i.d." assumption implies that the probability of flipping all heads over formula_25 flips is simply formula_26. Letting formula_27 yields 0, since formula_28 by assumption. The result is the same no matter how much we bias the coin towards heads, so long as we constrain formula_29 to be strictly between 0 and 1. In fact, the same result even holds in non-standard analysis—where infinitesimal probabilities are allowed.
Moreover, the event "the sequence of tosses contains at least one formula_30" will also happen almost surely (i.e., with probability 1).
But if instead of an infinite number of flips, flipping stops after some finite time, say 1,000,000 flips, then the probability of getting an all-heads sequence, formula_31, would no longer be 0, while the probability of getting at least one tails, formula_32, would no longer be 1 (i.e., the event is no longer almost sure).
Asymptotically almost surely.
In asymptotic analysis, a property is said to hold "asymptotically almost surely" (a.a.s.) if over a sequence of sets, the probability converges to 1. This is equivalent to convergence in probability. For instance, in number theory, a large number is asymptotically almost surely composite, by the prime number theorem; and in random graph theory, the statement "formula_33 is connected" (where formula_34 denotes the graphs on formula_25 vertices with edge probability formula_29) is true a.a.s. when, for some formula_37
formula_38
In number theory, this is referred to as "almost all", as in "almost all numbers are composite". Similarly, in graph theory, this is sometimes referred to as "almost surely".
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Time and Chance
Time and Chance may refer to:
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Chance Encounter
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Frequency (disambiguation)
Frequency refers to how often an event occurs within a given period.
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L'Événement
L'Événement means "The Happening" in French. It may refer to:
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Chances
Chances may refer to:
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Chance Operations (disambiguation)
Chance Operations or chance operations may refer to:
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Chance medley
Chance medley (from the Anglo-French "chance-medlee", a mixed chance), also 'chaunce medley' or 'chaude melle', is a term from English law used to describe a homicide arising from a sudden quarrel or fight. In other words, the term describes "the casual killing of a man, not altogether without the killer's fault, though without an evil intent; homicide by misadventure". The term distinguishes a killing that lacks malice aforethought necessary for murder, on the one hand, and pure accident on the other.
An early version of voluntary manslaughter, "chance medley" was a common defense in the 16th and 17th centuries but had fallen out of use by the 18th century, gradually replaced by the doctrine of provocation.
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Chance of a Lifetime
Chance of a Lifetime or The Chance of a Lifetime may refer to:
See also.
Topics referred to by the same