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Revision as of 19:19, 11 January 2003

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.

The idea is most often broken into two concepts:

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
epistemic probability, which represents our uncertainty of belief about propositions of which one does not know whether they are true. Such propositions may be about past or future events, but need not be.

Examples of epistemic probability:

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. The same mathematical rules apply regardless of what interpretation you favor.

Probability in mathematics

In probability theory, the probability $P$ of an event $x_{i}$ in a set of elementary events $S={x_{1},x_{2},...,x_{M}}$ , is the number of occurrances $N(x_{i})$ of that event divided by all possible events:

P(x_{i})={N(x_{i}) \over \sum _{j=1}^{N}N(x_{j})}.

As you can see from this formula, $P(x_{i})$ is a real number limited to the interval $[0, 1]$ .

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 occurrance 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 occurrances 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 occurrance of two events is related.

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