But in fact all the probabilities are equally real, depending on your selection process.

This is not so.
You are confused between two kinds of uncertainty (and so, probability): the uncertainty of the actual outcome in the real, physical world, and the uncertainty of some agent **not knowing** the outcome.

For a random person, the total probability of getting cancer will be 45.5%.

Let's unroll this. The actual probability for a random person to get cancer is either 90% or 1%. You just don't know which one of these two numbers applies, so you produce an estimate by combining them. Your estimate doesn't change anything in the real world and someone else -- e.g. someone who has access to the lesion-scanning results for this random person -- would have a different estimate.

Note, by the way, the difference between speaking about a "random person" and about the whole population. For the population as a whole, the 45.5% value is correct: out of 1000 people, about 455 will get cancer. But for a single person it is not correct: a single person has either a 90% **actual** probability or a 1% **actual** probability.

For simplicity consider an urn containing an equal number of white and black balls. You would say that a "random ball" has a 50% chance of being black -- but each ball is either black or white, it's not 50% of anything. 50% of the entire set of balls is black, true, but each ball's state is not uncertain and is not subject to ("actual") probability.

*1 point [-]