Editing Probability interpretations (section)

==Classical definition==
{{Main|Classical definition of probability}}
The first attempt at mathematical rigour in the field of probability, championed by [[Pierre-Simon Laplace]], is now known as the '''classical definition'''. Developed from studies of games of chance (such as rolling [[dice]]) it states that probability is shared equally between all the possible outcomes, provided these outcomes can be deemed equally likely.<ref name=SEPIP /> (3.1)

{{Quotation|The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.|Pierre-Simon Laplace|A Philosophical Essay on Probabilities<ref name=LaPlace>Laplace, P. S., 1814, English edition 1951, A Philosophical Essay on Probabilities, New York: Dover Publications Inc.</ref>}}
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[[Image:Dice.jpg|thumb|180px|right|The classical definition of probability works well for situations with only a finite number of equally-likely outcomes.]]
This can be represented mathematically as follows:
If a random experiment can result in ''N'' mutually exclusive and equally likely outcomes and if ''N<sub>A</sub>'' of these outcomes result in the occurrence of the event ''A'', the '''probability of ''A''''' is defined by
:<math>P(A) = {N_A \over N}. </math>

There are two clear limitations to the classical definition.<ref name="Spanos">{{cite book | last = Spanos | first = Aris | title = Statistical foundations of econometric modelling | publisher = Cambridge University Press | location = Cambridge New York | year = 1986 | isbn = 978-0521269124 }}</ref> Firstly, it is applicable only to situations in which there is only a 'finite' number of possible outcomes. But some important random experiments, such as [[Coin flipping|tossing a coin]] until it shows heads, give rise to an [[Infinity|infinite]] set of outcomes. And secondly, it requires an a priori determination that all possible outcomes are equally likely without falling in a trap of [[circular reasoning]] by relying on the notion of probability. (In using the terminology "we may be equally undecided", Laplace assumed, by what has been called the "[[principle of insufficient reason]]", that all possible outcomes are equally likely if there is no known reason to assume otherwise, for which there is no obvious justification.<ref>{{cite book |title=Decision Behaviour, Analysis and Support |author=Simon French |author2=John Maule |author3=Nadia Papamichail |publisher=Cambridge University Press |year=2009 |isbn=978-1-139-48098-7 |url=https://books.google.com/books?id=K-eMAgAAQBAJ&dq=%22principle+of+insufficient+reason%22&pg=PA221 |page=221}}</ref><ref>{{cite book |title=Philosophy of Probability |author=Nils-Eric Sahlin |editor=J. P. Dubucs |publisher=Springer |year=2013 |isbn=978-94-015-8208-7 |chapter-url=https://books.google.com/books?id=8djyCAAAQBAJ&dq=%22equally+likely%22+%22no+obvious+justification%22&pg=PA30 |page=30 |chapter=2. On Higher Order Beliefs}}</ref>)