Related concepts and fundamentals: *
* v * t * e PROBABILITY is the measure of the likelihood that an event will
occur.
These concepts have been given an axiomatic mathematical
formalization in probability theory , which is used widely in such
areas of study as mathematics , statistics , finance , gambling ,
science (in particular physics ), artificial intelligence /machine
learning , computer science , game theory , and philosophy to, for
example, draw inferences about the expected frequency of events.
CONTENTS * 1 Interpretations
* 2
* 6 Mathematical treatment * 6.1 Independent events
* 6.2
* 7 Relation to randomness and probability in quantum mechanics * 8 See also * 9 Notes * 10 Bibliography * 11 External links INTERPRETATIONS Main article:
When dealing with experiments that are random and well-defined in a purely theoretical setting (like tossing a fair coin), probabilities can be numerically described by the number of desired outcomes divided by the total number of all outcomes. For example, tossing a fair 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 1/4 or 0.25 (or 25%). When it comes to practical application however, there are two major competing categories of PROBABILITY INTERPRETATIONS, whose adherents possess different views about the fundamental nature of probability: * Objectivists assign numbers to describe some objective or physical
state of affairs. The most popular version of objective probability is
frequentist probability , which claims that the probability of a
random event denotes the _relative frequency of occurrence_ of an
experiment's outcome, when repeating the experiment. This
interpretation considers probability to be the relative frequency "in
the long run" of outcomes. A modification of this is propensity
probability , which interprets probability as the tendency of some
experiment to yield a certain outcome, even if it is performed only
once.
* Subjectivists assign numbers per subjective probability, i.e., as
a degree of belief. The degree of belief has been interpreted as,
"the price at which you would buy or sell a bet that pays 1 unit of
utility if E, 0 if not E." The most popular version of subjective
probability is
ETYMOLOGY See also:
The word _probability_ derives from the Latin _probabilitas_, which
can also mean "probity", a measure of the authority of a witness in a
legal case in
HISTORY Main article:
The scientific study of probability is a modern development of
mathematics.
According to
The sixteenth century Italian polymath
The theory of errors may be traced back to
The first two laws of error that were proposed both originated with
where h {displaystyle h} _ is a constant depending on
precision of observation, and c {displaystyle c} is a scale
factor ensuring that the area under the curve equals 1. He gave two
proofs, the second being essentially the same as
In the nineteenth century authors on the general theory included
On the geometric side (see integral geometry ) contributors to _The
Educational Times _ were influential (Miller, Crofton, McColl,
Wolstenholme, Watson, and
THEORY Main article:
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 interpreted or translated back into the problem domain. There have been at least two successful attempts to formalize
probability, namely the
There are other methods for quantifying uncertainty, such as the
APPLICATIONS
A good 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 very 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.
The discovery of rigorous methods to assess and combine probability assessments has changed society. It is important for most citizens to understand how probability assessments are made, and how they contribute to decisions. 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 See also:
Consider an experiment that can produce a number of results. The collection of all possible results is called the sample space of the experiment. The power set of the sample space is formed by considering all different collections of possible results. For example, rolling a dice can produce six possible results. One collection of possible results gives an odd number on the dice. 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 dice 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 if you look at a collection of mutually exclusive events (events with no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), 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 P ( A ) {displaystyle P(A)} , p ( A ) {displaystyle p(A)} , or Pr ( A ) {displaystyle {text{Pr}}(A)} . 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 (that
is, the event of _A_ not occurring), often denoted as A , A
, A {displaystyle {overline {A}},A^{complement },neg A} _, or
A {displaystyle {sim }A} ; 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 1 6
= 5 6 {displaystyle =1-{tfrac {1}{6}}={tfrac {5}{6}}} . See
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 P ( A B ) {displaystyle P(Acap B)} . INDEPENDENT EVENTS If two events, _A_ and _B_ are independent then the joint probability is P ( A and B ) = P ( A B ) = P ( A ) P ( B ) , {displaystyle P(A{mbox{ and }}B)=P(Acap B)=P(A)P(B),,} for example, if two coins are flipped the chance of both being heads is 1 2 1 2 = 1 4 {displaystyle {tfrac {1}{2}}times {tfrac {1}{2}}={tfrac {1}{4}}} . MUTUALLY EXCLUSIVE EVENTS If either event _A_ or event _B_ occurs on a single performance of an experiment this is called the union of the events _A_ and _B_ denoted as P ( A B ) {displaystyle P(Acup B)} . If two events are mutually exclusive then the probability of either occurring is P ( A or B ) = P ( A B ) = P ( A ) + P ( B ) . {displaystyle P(A{mbox{ or }}B)=P(Acup B)=P(A)+P(B).} For example, the chance of rolling a 1 or 2 on a six-sided die is P ( 1 or 2 ) = P ( 1 ) + P ( 2 ) = 1 6 + 1 6 = 1 3 . {displaystyle P(1{mbox{ or }}2)=P(1)+P(2)={tfrac {1}{6}}+{tfrac {1}{6}}={tfrac {1}{3}}.} NOT MUTUALLY EXCLUSIVE EVENTS If the events are not mutually exclusive then P ( A or B ) = P ( A ) + P ( B ) P ( A and B ) . {displaystyle Pleft(A{hbox{ or }}Bright)=Pleft(Aright)+Pleft(Bright)-Pleft(A{mbox{ and }}Bright).} For example, when drawing a single card at random from a regular deck of cards, the chance of getting a heart or a face card (J,Q,K) (or one that is both) is 13 52 + 12 52 3 52 = 11 26 {displaystyle {tfrac {13}{52}}+{tfrac {12}{52}}-{tfrac {3}{52}}={tfrac {11}{26}}} , because of 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. CONDITIONAL PROBABILITY _
If P ( B ) = 0 {displaystyle P(B)=0} _ then P ( A B ) {displaystyle P(Amid B)} is formally undefined by this expression. However, it is possible to define a conditional probability for some zero-probability events 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 1 / 2 {displaystyle 1/2} ; 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, such as, if a red ball was taken, the probability of picking a red ball again would be 1 / 3 {displaystyle 1/3} since only 1 red and 2 blue balls would have been remaining. INVERSE PROBABILITY In probability theory and applications, BAYES\' RULE relates the odds
of event A 1 {displaystyle A_{1}} to event A 2
{displaystyle A_{2}} , before (prior to) and after (posterior to)
conditioning on another event B {displaystyle B} . The odds on
A 1 {displaystyle A_{1}} to event A 2
{displaystyle A_{2}} is simply the ratio of the probabilities of the
two events. When arbitrarily many events A {displaystyle A}
are of interest, not just two, the rule can be rephrased as POSTERIOR
IS PROPORTIONAL TO PRIOR TIMES LIKELIHOOD, P ( A B ) P ( A )
P ( B A ) {displaystyle P(AB)propto P(A)P(BA)} where the
proportionality symbol means that the left hand side is proportional
to (i.e., equals a constant times) the right hand side as A
{displaystyle A} varies, for fixed or given B {displaystyle B}
(Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to
SUMMARY OF PROBABILITIES Summary of probabilities EVENT PROBABILITY A P ( A ) {displaystyle P(A)in ,} not A P ( A ) = 1 P ( A ) {displaystyle P(A^{complement })=1-P(A),} A or B P ( A B ) = P ( A ) + P ( B ) P ( A B ) P ( A B ) = P ( A ) + P ( B ) if A and B are mutually exclusive {displaystyle {begin{aligned}P(Acup B)&=P(A)+P(B)-P(Acap B)\P(Acup B) width:65.295ex; height:6.176ex;" alt="{begin{aligned}P(Acup B)&=P(A)+P(B)-P(Acap B)\P(Acup B)"> P ( A B ) = P ( A B ) P ( B ) = P ( B A ) P ( A ) P ( A B ) = P ( A ) P ( B ) if A and B are independent {displaystyle {begin{aligned}P(Acap B)&=P(AB)P(B)=P(BA)P(A)\P(Acap B) width:56.041ex; height:6.176ex;" alt="{begin{aligned}P(Acap B)&=P(AB)P(B)=P(BA)P(A)\P(Acap B)"> P ( A B ) = P ( A B ) P ( B ) = P ( B A ) P ( A ) P ( B ) {displaystyle P(Amid B)={frac {P(Acap B)}{P(B)}}={frac {P(BA)P(A)}{P(B)}},} RELATION TO RANDOMNESS AND PROBABILITY IN QUANTUM MECHANICS Main article:
_ THIS SECTION NEEDS EXPANSION. You can help by adding to it . (April 2017)_ 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). Of course, 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
SEE ALSO *
Main article:
*
In
NOTES * ^ "Probability". _Webster\'s Revised Unabridged Dictionary _. G &
C Merriam, 1913
* ^ Strictly speaking, a probability of 0 indicates that an event
_almost_ never takes place, whereas a probability of 1 indicates than
an event _almost_ certainly takes place. This is an important
distinction when the sample space is infinite. For example, for the
uniform distribution on the real interval , there are an infinite
number of possible outcomes, and the probability of any given outcome
being observed — for instance, exactly 7 — is 0. This means that
when we make an observation, it will _almost surely not_ be exactly 7.
However, it does NOT mean that exactly 7 is _impossible_. Ultimately
some specific outcome (with probability 0) will be observed, and one
possibility for that specific outcome is exactly 7.
* ^ "Kendall's Advanced
BIBLIOGRAPHY * Kallenberg, O. (2005) _Probabilistic Symmetries and Invariance Principles_. Springer -Verlag, New York. 510 pp. ISBN 0-387-25115-4 * Kallenberg, O. (2002) _Foundations of Modern Probability,_ 2nd ed. Springer Series in Statistics. 650 pp. ISBN 0-387-95313-2 * Olofsson, Peter (2005) _Probability, Statistics, and Stochastic Processes_, Wiley-Interscience. 504 pp ISBN 0-471-67969-0 . EXTERNAL LINKS _ Wikiquote has |