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I have recently been thinking about the fundamental meaning of the term probability, so I decide to discuss the topic with my tutor. He told me that the true definition of the probability of x occuring, P(x), is:

P(x) = Lim(Relative frequency of x in experiments) as n tends to infinity, where n = the sample size.

However, I have read many mathemetics text books which talk about "experimental probabilities" and "theoretical probabilities" - they refer to the definition I have previously mentioned as "experimental probabilities". "Theoretical probabilities" are apparently defined as follows:

P(x) = (Number of ways event can occur)/(Total number of events which can occur)

I wish to know if there is any thorough algebriac way to proove that the fraction calculated through the "theoretical probability" method is infact the number which the relative frequency will converge to as the sample size gets ever larger?

Also, I was curious as to what we would call the probability of any event for which the limit of the relative frequency does not converge?

Finally, is it possible to calculate the value which the limit of relative frequency will take, or can this only be obtained through repeat experiment?

Many thanks - eagerly awaiting replies.

Simon.

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# Meaning and derivation of probability

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