Wikipedia: Probability

From the Latin probare (to prove, or to test), the word probable means roughly "likely to occur" in the case of possible future occurences, or "likely to be true" in the case of inferences from evidence. While mathematicians and scientists all agree on how to calculate the probability of certain events and how to use those calculations in certain ways, there is considerable disagreement on what the numbers mean in reality. The idea is most often broken into two concepts: [aleatory probability]?, which represents the likelihood of future events whose occurence is governed by some random physical phenomenon like tossing dice or spinning a wheel; and [epistemic probability]?, which represents our uncertainty of belief about past events that either did or did not occur, or uncertainty about the causes of future events. The latter, for example, is what we are talking about when we say that it is "probable" that a certain 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.

In probability theory, the basic elements are a set of elementary events, and a random variable (function) mapping the occurrence of each event in the sample space of events to the interval [0,1]. The probability that an event occurs is expressed as a real number in the interval [0,1] (inclusive). 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).

For an amusing probability riddle, see the Monty Hall problem.

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