What makes it legitimate to say a set of values are probabilities? (I need an answer for a formalisation reasons.)
Let me give a simple example. Let's assume that I have a number of letters and each letter has a value associated with it:
A : 0.2
B : 0.5
C : 0.9
D : 0.2
Is it enough divide the value of a letter by the sum these values to say that the probability the letter C for instance is 0.9/ (0.2+0.5+0.9+0.2) = 0.5?
What properties need to exist to call such a scenario a "probability"?
First off, there are basic probability axioms: you have a sample space of all possible observable values that arise from conducting an experiment, like the face of a thrown die. Put loosely, the probability of events must sum to 1 (among some other considerations). Standardizing (summing over some weights, and dividing by the sum) is a frequently used way of calculating probabilities.
In your experiment, are events A, B, C, and D mutually exclusive in the sense that your sample space is {A, B, C, D}? If the weights represent an unstandardized marginal frequencies, standardization is fine. Or is each calculated sequentially so you observe {{}, A, AB, ABC, ABCD, B, BC, BCD, C, CD, D} possibly and the frequencies correspond to marginal probabilities? In that case, C might really show up in 90% of trials and B in 50%. (if these were mutually exclusive outcomes, that would violate probability axioms since their probability sum is 140%).
Second, the question "what makes a probability" is philosophical in nature. Bayesians take probability to mean a degree of belief whereas frequentists take it to mean a frequency in the sense of a thought experiment involving independent replications of a particular experiment or data collection procedure.
