Contents

Probability

Probability derives from the Latin probare (to prove, or to test). The word probable means roughly "likely to occur" in the case of possible future occurrences, or "likely to be true" in the case of inferences from evidence. See also probability theory

What mathematicians call probability is the mathematical theory we use to describe and quantify uncertainty. In a larger context, (see probability interpretations) the word probability is used with other concerns in mind. Uncertainty can be due to our ignorance, deliberate mixing or shuffling, or due to the essential randomness of Nature. In any case, we measure the uncertainty of events on a scale from zero (impossible events) to one (certain events or no uncertainty).

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

The idea is most often broken into two concepts:

1. aleatory probability, which represents the likelihood of future events whose occurrence is governed by some random physical phenomenon like tossing dice or spinning a wheel; and
2. epistemic probability, which represents our uncertainty about propositions when one lacks complete knowledge of causitive circumstances. Such propositions may be about past or future events, but need not be. Some examples of epistemic probability are:
• Assign a probability to the proposition that a proposed law of physics is true.
• Determine how "probable" it is that a suspect committed a crime, based on the evidence presented.

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

 Probability in mathematics

While the existence of gambling games of chance shows that there has been a lively interest in quantifying the ideas of probability for millenia, exact mathematical descriptions of use in these types of problems only arose much later.

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

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

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

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

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

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

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

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

One contribution of Bayesian probability was to provide a philosophical stance which allows us to derive probabilities (according to the above definition) from a set of observations.

 Formalization

In general, probabilities of interest regard not just discrete outcomes like "heads/tails", but also more continuous outcomes as well.

In probability theory, an event is a "measurable" subset of a "sample space". "Events" are the things to which probabilities are assigned. A probability is a number in the closed interval from 0 to 1. Probabilities must be assigned to events in such a way that for pairwise disjoint (i.e., no two intersect each other) events A1, A2, A3, ..., the probability of their union is the sum of their probabilities, or, in mathematical notation,

$P\left(A_1\cup A_2\cup A_3\cup\cdots\right) =P(A_1)+P(A_2)+P(A_3)+\cdots.$ In the special case of a "discrete probability distribution" the sample space is a set $\left\{\,x_1,x_2,x_3,...\,\right\}$ of outcomes to each of which a positive number has been assigned as its probability. The one-members sets $\{\,x_i\,\}$ are "elementary events". One of the simplest of discrete sample spaces is a finite set $\{\,x_1,x_2,x_3,...,x_n\,\}$ to each of whose members the same probability 1/n is assigned. An example of a sample space that is not discrete is the closed interval [0, 1] to which the length of any subinterval (a, b) is assigned as the probability of that subinterval. The probability assigned to any one-member subset is 0.

### Representation and interpretation of probability values

The value 0 is generally understood to represent impossible events, while the number 1 is understood to represent certain events (though there are more advanced interpretations of probability that use more precise definitions). Values between 0 and 1 quantify the probability of the occurrence of some event. In common language, these numbers are often expressed as fractions or percentages, and must be converted to real number form to perform calculations with them.

For example, if two events are equally likely, such as a flipped coin landing heads-up or tails-up, we express the probability of each event as "1 in 2" or "50%" or "1/2", where the numerator of the fraction is the relative likelihood of the target event and the denominator is the total of relative likelihoods for all events. To use the probability in math we must perform the division and express it as "0.5".

Another way probabilities are expressed is "odds", where the two numbers used represent the relative likelihood of the target event and the likelihood of all events other than the target event. Expressed as odds, tossing a coin will give heads odds of "1 to 1"or "1:1". To convert odds to probability, use the sum of the numbers given as the denominator of a fraction: "1:1" odds make a "1/2" probability; "3:2" odds make a "3/5" probability (or 0.6).

### Distributions

The histogram of events versus occurrence is called a probability distribution. There are several important, discrete distributions, such as the discrete uniform distribution, the Poisson distribution, the binomial distribution, the negative binomial distribution and the Maxwell-Boltzmann distribution.

### Remarks on probability calculations

The difficulty of probability calculations lie in determining the number of possible events, counting the occurrences of each event, counting the total number of possible events. Especially difficult is drawing meaningful conclusions from the probabilities calculated. An amusing probability riddle, the Monty Hall problem demonstrates the pitfalls nicely.

To learn more about the basics of probability theory, see the article on probability axioms and the article on Bayes' theorem that explains the use of conditional probabilities in case where the occurrence of two events is related.

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

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

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

• Damon Runyon, "It may be that the race is not always to the swift, nor the battle to the strong - but that is the way to bet."