Estimate unknown \(f\) from paired data so we can predict; physics of \(f\) is often unavailable, so we use repeated observations + probability.
Set \(\Omega\) of all possible outcomes of a chosen random experiment; no, it need not be numeric (e.g. \(\{H,T\}\)).
e.g. single toss vs five tosses — different experiments \(\to\) different \(\Omega\).
The whole pipeline: illness / visit \(\to\) imaging \(\to\) digital image \(\to\) clinician label (as one generative process whose outcomes we observe).
To compare / quantify subsets (events); a bare set does not tell you “how much” or “how likely.”
Non-negative; \(P(\Omega)=1\); \(P(\emptyset)=0\); additive on disjoint events; values in \([0,1]\).
A subset of \(\Omega\) (in the collection \(\mathcal{F}\)) that we assign probability to.
The math works with \(P\) as an abstract measure; human/clinical meaning is attached when you must make a real-world decision.