Blackwell Reference Online

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Philosophical theories of the meaning of “probability” have dealt with a variety of concepts which may be loosely grouped into three families: degree of belief , relative frequency and chance . The mathematics of probability has been less controversial than its interpretation, largely because all theories treat probability as some kind of proportion. Frequentists treat probability as the proportion of actual cases displaying an attribute. Degree of belief and chance theories treat probability as a proportion of possible cases, with the measure on which the proportions are based reflecting respectively intensities of belief that one has or ought to have or intensities of causal tendency. Proportions add. If I have one half of the pie and you have one quarter of it (and we aren't sharing any) then together we have three quarters of the pie. Proportions are normalized. All of the pie is 100 per cent. Proportions can't be negative. Formally, we can think of a probability as a function mapping elements of a Boolean algebra B (possible pie slices including the null slice and the whole pie) to real numbers in the interval [0, 1] which satisfies: 1.  If b, c are disjoint then pr(b ∪ c) = pr(b) + pr(c) 2.  pr (universal element) = 1 3.  For no b, is pr(b) < o. (It is often assumed that there is an underlying space (the set of pie atoms) such that the Boolean algebra in the foregoing ... log in or subscribe to read full text