Speaking of probability measure (a number from 0 to 1), textbooks usually describe several interpretations of this number. E.g., Practical Business Statistics by Andrew F. Siegel reads (p. 132):
Probability numbers can come from previous experience (relative frequency as a percentage), from a mathematical statement (theoretical probability), or they can be simply made up (subjective probability, yes it is permitted to use someone’s opinion here).
For my question, the third interpretation is redundant, so I will consider the first two. The question itself is: Do these two interpretations correspond to any single thing at all?
I will try to explain my doubts in detail. Suppose we have two events: the first event’s probability was calculated theoretically (let the event be something like random card picking), and the second event’s probability was assessed on a frequency basis—like probability, say, of having a specific number of offsprings in a litter, estimated by numerous observations of this species’ reproduction in the past. So, is there any evidence that these two probabilities belong to the same category? They are both measures of uncertainity of some kind, but what makes us believe that it is a single kind and not two different ones? E.g. the luminous intensity and the luminance both measure the light in one way or another, but they are two different measures, and you cannot directly compare what is bigger—5 candela or 10 lux.
So why are we sure that theoretical probability and frequency both refer to the same quantity? What are the reasons for comparing like “to pick the queen of spades is twice as likely as to have eight offsprings in a litter”? For me, it sounds rather like “the height of Mount Fuji is twice the average speed of a snail”.
What am I missing, or what do I musunderstand?