Say I have eaten hamburgers every Tuesday for years. You could say that I eat hamburgers 14% of the time, or that the probability of me eating a hamburger in a given week is 14%.
What are the main differences between probabilities and proportions?
Is a probability an expected proportion?
Are probabilities uncertain and proportions are guaranteed?
Best Answer
I have hesitated to wade into this discussion, but because it seems to have gotten sidetracked over a trivial issue concerning how to express numbers, maybe it's worthwhile refocusing it. A point of departure for your consideration is this:
A frequentist might rely on laws of large numbers to justify statements like "the long-run proportion of an event [is] its probability." This supplies meaning to statements like "a probability is an expected proportion," which otherwise might appear merely tautological. Other interpretations of probability also lead to connections between probabilities and proportions but they are less direct than this one.
In our models we usually take probabilities to be definite but unknown. Due to the sharp contrasts among the meanings of "probable," "definite," and "unknown" I am reluctant to apply the term "uncertain" to describe that situation. However, before we conduct a sequence of observations, the [eventual] proportion, like any future event, is indeed "uncertain". After we make those observations, the proportion is both definite and known. (Perhaps this is what is meant by "guaranteed" in the OP.) Much of our knowledge about the [hypothetical] probability is mediated through these uncertain observations and informed by the idea that they might have turned out otherwise. In this sense--that uncertainty about the observations is transmitted back to uncertain knowledge of the underlying probability--it seems justifiable to refer to the probability as "uncertain."
In any event it is apparent that probabilities and proportions function differently in statistics, despite their similarities and intimate relationships. It would be a mistake to take them to be the same thing.
Reference
Huber, WA Ignorance is Not Probability. Risk Analysis Volume 30, Issue 3, pages 371–376, March 2010.