Probability 

Part of a series on 
Certainty 

Related concepts and fundamentals: 
Probability is the measure of the likelihood that an event will occur.^{[1]} See glossary of probability and statistics. Probability quantifies as a number between 0 and 1, where, loosely speaking,^{[note 1]} 0 indicates impossibility and 1 indicates certainty.^{[2]}^{[3]} The higher the probability of an event, the more likely it is that the event will occur. 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 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. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.^{[4]}
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✪ Intro to Conditional Probability
Transcription
Hi I’m Rob, welcome to Math Antics! In this video we’re gonna learn about how to do math with things that only sometimes happen. They might be likely or unlikely. We’re gonna learn about Probability. Usually in math we deal with things that always happen the same way. They’re completely certain. Like if you add 1 and 1 you’re always gonna get 2. If you multiply 2 and 3 you’re always gonna get 6. There’s no uncertainty at all. But in the real world things aren’t always so predictable. Take a coin toss for example. We can’t predict whether it will be heads or tails. It’s unpredictable or random, and that’s why some people will flip a coin to help decide which of two things to do. That’s how I make every decision in life Why am I not surprised? Oh no! [sound of thunder] Good luck But even though we don’t know what each coin flip is going to be, we do know a few things about it. We know that with a fair coin toss that heads is just as likely to show up as tails. The Probability of an event (like getting heads or getting tails) is a value that tells us HOW LIKELY that event is to happen. Oh, and sometimes instead of saying “the probability of an event” people will say the “odds” or “chances” of an event which means the same thing. With our coin toss, since each side is just as likely, and there’s only two sides to a coin, if we flipped a coin a lot of times, we should expect that about half the flips will be heads and about half the flips will be tails. And that means the probability of flipping heads is the fraction onehalf and the probability of flipping tails is also onehalf. Let’s look at this in a little more detail on something called a Probability Line. It’s a number line that goes from 0 to 1. A probability of zero means that an event cannot happen, it’s impossible. And a probability of 1 means that an event is definitely going to happen, it’s certain. That’s why the probability line only goes from 0 to 1. An event can’t be less likely than impossible and it can’t be more likely than certain. A probability of onehalf (like with our coin toss) means that an event is just as likely to happen as it is to NOT happen. A probability less than onehalf means that an event is unlikely and a probability greater than onehalf means that an event is likely. Oh, and in addition to fractions, it’s also common to write probabilities as decimals or percentages, since you can easily convert between those three. A probability of 0 is the same as a 0 percent chance of something happening, a probability of onehalf is the same as a 50 percent chance of something happening, and a probability of 1 is the same as a 100 percent chance of something happening. Now that you know how a coin toss works, lets see an example of an event that is unlikely using something a little more complicated than a coin. Let’s take a look at dice. A standard die has 6 sides numbered 1 through 6. When you roll it, any of those sides is just as likely to come up as the others. That sounds a lot like flipping a coin doesn’t it? Each side of a die is just as likely to come up as the others and each side of a coin was just as likely to come up as the others. So you might expect that the likelihood of rolling a 3 is 50%. But remember, with a coin toss there were only 2 possibilities: heads or tails. With dice there are 6 possibilities. And that’s going to make a difference in its probability. One way to think about it is that it’s certain that one of those six sides will land facing upwards, which is a probability of 1 (or 100%) But since ONLY ONE side can face upwards for a given roll, we have to divide up that value among all the possibilities. In the case of a coin toss, since there were only 2 possibilities, we had to divide the probability by 2. 1 divided by 2 is onehalf, (which is the decimal 0.5 or 50%) But with the die, we need to divide the probability up evenly between 6 possibilities. 1 divided by 6 is onesixth which is equivalent to 0.167 (or 16.7%) So that would be right here on our probability line. That means it isn’t likely that I would roll a 3 for instance, but it’s just as likely as rolling any other number. And since all 6 numbers have the same probability, each number should come up about as often as the others. To see if they do, I’m going to conduct some trials. That’s an excellent argument. Allow me to deliberate. Guilty! Actually, when dealing with probability, a trial (which can also be called an experiment) is a process that has a random outcome …like tossing a coin or rolling dice or spinning a spinner. And the outcome of a trial is what happens in that particular trial. Like flipping heads, or rolling a 3. So I’m gonna conduct several trials by rolling a die multiple times and keeping track of how many times I roll each number. Ah ha! You said that each number was gonna come up just as often as the other numbers. But look! There’s more 2s than there are 5s. How do you explain that?! Well remember, we’re dealing with things that are random. They’re unpredictable. We can’t know exactly what will happen, just what will happen on average. So now I have to calculate the average? Well, when we say “on average” we mean that the more trials you do the closer you get to the expected probabilities. Keep watching. There. Now that we’ve done a LOT of trials you can see that our totals are much closer to what we would expect them to be. I guess you’re probably right. That’s one of the really important things to keep in mind about probability. If you do just a few trials, the results might not end up very close to what you’d expect. In fact, they could be way off! But the more you try, the closer they will match the expected probabilities. And there’s another thing I should point out... Remember, the odds of flipping heads is 1/2 and the odds of flipping tails is 1/2. The odds of rolling a 1 is 1/6, and the odds of rolling any other number on a die is 1/6. If you add up the probabilities for the coin flip, you get 2/2 or 1. And if you add up the probabilities for rolling a die, you get 6/6 which is also 1. And that’s not just a coincidence. If you add up the probabilities of all possible outcomes of a trial, the total is going to be 1 or 100% because it is certain that at least one of those possibilities will happen. Let’s look at some more examples. For these examples we’ll use a spinner. If we had a spinner with just six equally sized sectors, the probabilities would be exactly the same as with dice. So we want a few more sectors. There that’s more like it. Now we have 16 equally sized sectors. So, what is the probability of spinning a 12? Well, just like with the dice where we had to split up the 100% between all six possibilities, we’ll do the same thing now, but we’ll split it up between 16 possibilities. So the probability of spinning a 12 is 1/16 or about 6%. Which is right here on the probability line. We can see that the probability of spinning a 12 is less likely than the probability of rolling a 3. And that makes sense because there are more possible outcomes with our spinner. But what if we color some of the sectors a different color and we want to know the odds of spinning a certain color? Now we have 5 sectors colored blue and 11 sectors colored yellow. So what is the probability of spinning a blue? Remember how with a coin toss, we ended up with the fraction 1 over 2 and with a die roll we got the fraction 1 over 6. In both cases we had 1 as the numerator. And that’s because we were interested in only ONE of the possible outcomes, like the probability of flipping heads or the probability of the number 3 being rolled. But in this case the top number of our fraction will be 5 because any of these 5 sectors will give us the color we want. And the bottom number will still be the total number of possibilities, which is 16 because that’s how many total sectors we have. So the probability of spinning a blue is 5/16 or about 31%. That’s still considered unlikely, but it is more likely than spinning a specific number. And this method will work for figuring out the probability of any event. You just make a fraction with the numerator as the number of outcomes that satisfy your requirement. And the denominator as the total number of possible outcomes. Let’s try the same method to find out the odds of spinning a yellow. Our top number should be 11 because there’s 11 yellow sectors. And our bottom number should still be 16. So the odds of spinning a yellow are 11/16 or about 69%. Now we finally have a probability that’s considered likely. And it makes sense, because you can see by looking at our spinner that it’s more likely to spin a yellow than a blue. And you’ll notice, if we add up 5/16 and 11/16 we get 16/16 or a probability of 1. So that’s a good sign that we did it right. Let’s look at another example. Suppose we have a bag of marbles. There are 3 green marbles, 7 yellow marbles and 1 white marble. If we mix them up and pull out a marble at random, what’s the probability of it being green? Well, the top number of our probability fraction will be 3 because there’s 3 green marbles so there’s 3 outcomes that get us what we want. And the bottom number will be 11 because there’s a total of 11 possible marbles that we could pull out. So the probability of pulling out a green marble is 3/11 or 0.27 or 27% It’s right here on the Probability Line. That means it’s unlikely. And that makes sense because you can see that it would be less likely to pull out a green marble than one of the other ones. Let’s try this again for calculating the probability of pulling out a yellow marble. This time the numerator of our fraction will be 7 because there’s 7 yellow marbles. The denominator will still be 11 because there are still 11 marbles total. So the probability of pulling out a yellow marble is 7/11 or 0.64 or 64%. …another example of an event that is likely. …how about pulling out the white marble? Well, the top number will be 1 since there’s only one white marble. And the bottom number is still 11. So the probability of pulling out a white marble is 1/11 or 0.09 or 9% …not very likely. And if we add up these probabilities we get 11/11, or 100%, just as we expected. All right! So you should have a pretty good handle on basic probability now. You just have to remember to make a fraction with the numerator being the number of outcomes that give you what you want, and the denominator being the total number of possibilities. And we learned about the Probability Line, and that a probability can’t be less than 0 or greater that 1 (or 100%). We also learned that the more trials or experiments you conduct, the closer your results will get to the expected probabilities. Of course the way to get good at it is to practice. So be sure to do a lot of problems on your own. As always, thanks for watching Math Antics and I’ll see ya next time. And I sentence you to… ...five years hard labor! Learn more at www.mathantics.com
Contents
Interpretations
When dealing with experiments that are random and welldefined 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 "headhead", "headtail", "tailhead", and "tailtail" outcomes. The probability of getting an outcome of "headhead" 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 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.^{[5]} 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.^{[6]} 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."^{[7]} The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as experimental data to produce probabilities. The expert knowledge is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a posterior probability distribution that incorporates all the information known to date.^{[8]} By Aumann's agreement theorem, Bayesian agents whose prior beliefs are similar will end up with similar posterior beliefs. However, sufficiently different priors can lead to different conclusions regardless of how much information the agents share.^{[9]}
Etymology
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 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.^{[10]}
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 for millennia, 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^{[clarification needed]} are still obscured by the superstitions of gamblers.^{[11]}
According to Richard Jeffrey, "Before the middle of the seventeenth century, the term 'probable' (Latin probabilis) meant approvable, and was applied in that sense, unequivocally, to opinion and to action. A probable action or opinion was one such as sensible people would undertake or hold, in the circumstances."^{[12]} However, in legal contexts especially, 'probable' could also apply to propositions for which there was good evidence.^{[13]}
Forms of probability and statistics were developed by Arab mathematicians studying cryptology between the 8th and 13th centuries. AlKhalil (717–786) wrote the Book of Cryptographic Messages which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. AlKindi (801–873) made the earliest known use of statistical inference in his work on cryptanalysis and frequency analysis. An important contribution of Ibn Adlan (1187–1268) was on sample size for use of frequency analysis.^{[14]}
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^{[15]}). 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.^{[16]} Jakob Bernoulli's Ars Conjectandi (posthumous, 1713) and Abraham de Moivre's Doctrine of Chances (1718) treated the subject as a branch of mathematics.^{[17]} See Ian Hacking's The Emergence of Probability^{[10]} and James Franklin's The Science of Conjecture^{[18]} 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.^{[19]} 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 PierreSimon 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.^{[20]} 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 wellknown precocity had probably not made this discovery before he was two years old."^{[20]}
Daniel Bernoulli (1778) introduced the principle of the maximum product of the probabilities of a system of concurrent errors.
AdrienMarie 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).^{[21]} In ignorance of Legendre's contribution, an IrishAmerican writer, Robert Adrain, editor of "The Analyst" (1808), first deduced the law of facility of error,
where is a constant depending on precision of observation, and 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).^{[citation needed]} 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^{[clarification needed]} 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.
Andrey Markov introduced^{[22]} the notion of Markov chains (1906), which played an important role in stochastic processes theory and its applications. The modern theory of probability based on the measure theory was developed by Andrey Kolmogorov (1931).^{[23]}
On the geometric side (see integral geometry) contributors to The Educational Times were influential (Miller, Crofton, McColl, Wolstenholme, Watson, and Artemas Martin).^{[citation needed]}
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 probability space), sets are interpreted as events and probability itself as a measure on a class of sets. In Cox's theorem, 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.
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 laws of probability as usually understood.
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 (Reliability theory of aging and longevity), and financial regulation.
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.^{[24]}
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.^{[25]}
The discovery of rigorous methods to assess and combine probability assessments has changed society.^{[26]}^{[citation needed]}
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.^{[27]}
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. 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.^{[28]}
The probability of an event A is written as , , or .^{[29]} 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 , or ; its probability is given by P(not A) = 1 − P(A).^{[30]} As an example, the chance of not rolling a six on a sixsided die is 1 – (chance of rolling a six) . See Complementary event for a more complete treatment.
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 .
Independent events
If two events, A and B are independent then the joint probability is
for example, if two coins are flipped the chance of both being heads is .^{[31]}
Mutually exclusive events
If either event A or event B but never both occurs on a single performance of an experiment, then they are called mutually exclusive events.
If two events are mutually exclusive then the probability of both occurring is denoted as .
If two events are mutually exclusive then the probability of either occurring is denoted as .
For example, the chance of rolling a 1 or 2 on a sixsided die is
Not mutually exclusive events
If the events are not mutually exclusive then
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 , 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
Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written , and is read "the probability of A, given B". It is defined by^{[32]}
If then is formally undefined by this expression. However, it is possible to define a conditional probability for some zeroprobability events using a σalgebra of such events (such as those arising from a continuous random variable).^{[citation needed]}
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 ; 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 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 to event , before (prior to) and after (posterior to) conditioning on another event . The odds on to event is simply the ratio of the probabilities of the two events. When arbitrarily many events are of interest, not just two, the rule can be rephrased as posterior is proportional to prior times likelihood, where the proportionality symbol means that the left hand side is proportional to (i.e., equals a constant times) the right hand side as varies, for fixed or given (Lee, 2012; Bertsch McGrayne, 2012). In this form it goes back to Laplace (1774) and to Cournot (1843); see Fienberg (2005). See Inverse probability and Bayes' rule.
Summary of probabilities
Event  Probability 

A  
not A  
A or B  
A and B  
A given B 
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 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 6.02×10^{23}) that only a statistical description of its properties is feasible.
Probability theory is required to describe quantum phenomena.^{[33]} A revolutionary discovery of early 20th century physics was the random character of all physical processes that occur at subatomic 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 remarked in a letter to Max Born: "I am convinced that God does not play dice".^{[34]} Like Einstein, Erwin Schrödinger, who discovered the wave function, believed quantum mechanics is a statistical approximation of an underlying deterministic reality.^{[35]} In some modern interpretations of the statistical mechanics of measurement, quantum decoherence is invoked to account for the appearance of subjectively probabilistic experimental outcomes.
See also
 Chance (disambiguation)
 Class membership probabilities
 Contingency
 Equiprobability
 Heuristics in judgment and decisionmaking
 Probability theory
 Randomness
 Statistics
 Estimators
 Estimation Theory
 Probability density function
 In Law
Notes
 ^ 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 continuous uniform distribution on the real interval [5, 10], 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.
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 ^ Singh, Laurie (2010) "Whither Efficient Markets? Efficient Market Theory and Behavioral Finance". The Finance Professionals' Post, 2010.
 ^ Gao, J.Z.; Fong, D.; Liu, X. (April 2011). "Mathematical analyses of casino rebate systems for VIP gambling". International Gambling Studies. 11 (1): 93–106. doi:10.1080/14459795.2011.552575.
 ^ "Data: Data Analysis, Probability and Statistics, and Graphing". archon.educ.kent.edu. Retrieved 28 May 2017.
 ^ Gorman, Michael (2011) "Management Insights". Management Science^{[full citation needed]}
 ^ Ross, Sheldon. A First course in Probability, 8th Edition. pp. 26–27.
 ^ Olofsson (2005) p. 8.
 ^ Olofsson (2005), p. 9
 ^ Olofsson (2005) p. 35.
 ^ Olofsson (2005) p. 29.
 ^ Burgi, Mark (2010) "Interpretations of Negative Probabilities", p. 1. arXiv:1008.1287v1
 ^ Jedenfalls bin ich überzeugt, daß der Alte nicht würfelt. Letter to Max Born, 4 December 1926, in: Einstein/Born Briefwechsel 1916–1955.
 ^ Moore, W.J. (1992). Schrödinger: Life and Thought. Cambridge University Press. p. 479. ISBN 9780521437677.
Bibliography
 Kallenberg, O. (2005) Probabilistic Symmetries and Invariance Principles. SpringerVerlag, New York. 510 pp. ISBN 0387251154
 Kallenberg, O. (2002) Foundations of Modern Probability, 2nd ed. Springer Series in Statistics. 650 pp. ISBN 0387953132
 Olofsson, Peter (2005) Probability, Statistics, and Stochastic Processes, WileyInterscience. 504 pp ISBN 0471679690.
External links
Wikiquote has quotations related to: Probability 
Wikibooks has more on the topic of: Probability 
Wikimedia Commons has media related to Probability. 
Library resources about Probability 
 Virtual Laboratories in Probability and Statistics (Univ. of Ala.Huntsville)
 Probability on In Our Time at the BBC
 Probability and Statistics EBook
 Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Preprint: Washington University, (1996). — HTML index with links to PostScript files and PDF (first three chapters)
 People from the History of Probability and Statistics (Univ. of Southampton)
 Probability and Statistics on the Earliest Uses Pages (Univ. of Southampton)
 Earliest Uses of Symbols in Probability and Statistics on Earliest Uses of Various Mathematical Symbols
 A tutorial on probability and Bayes' theorem devised for firstyear Oxford University students
 [1] pdf file of An Anthology of Chance Operations (1963) at UbuWeb
 Introduction to Probability – eBook, by Charles Grinstead, Laurie Snell Source (GNU Free Documentation License)
 (in English) (in Italian) Bruno de Finetti, Probabilità e induzione, Bologna, CLUEB, 1993. ISBN 8880911767 (digital version)
 Richard P. Feynman's Lecture on probability.