Probability and conditioning

A sample space collects the outcomes your model treats as possible. An event is a subset of those outcomes. Probability assigns nonnegative weights, with total weight one, and adds weights for disjoint events. For a fair six-sided die, every face receives probability one sixth. Fairness is a modeling assumption rather than a consequence of the die having six sides; a damaged die can use the same sample space with different weights.

Complements and overlaps let you calculate more complicated events. The probability of either event happening is the sum of their probabilities minus the overlap that was counted twice. Mutually exclusive events have no overlap. Independence means something different: knowing that one occurred does not alter the other's probability. Two positive-probability exclusive events cannot be independent, because observing one makes the other impossible.

The conditional probability $P(A\mid B)$ asks what fraction of the probability inside event $B$ also belongs to $A$. Dividing by $P(B)$ renormalizes that restricted population. Reversing the condition usually changes the answer: the fraction of faults that trigger an alert need not equal the fraction of alerts caused by faults. The denominators describe different populations even though the numerator involves the same overlap.

Bayes' rule relates these directions. Its denominator must account for every way the evidence can arise. In a fault detector, alerts can come from true faults and false positives. Ignoring the second route can create a wildly overconfident conclusion when faults are rare. Conditioning can also be undefined for a zero-probability event in this elementary formula; continuous conditioning requires more careful machinery introduced later.

Independent repeated trials make some models manageable, but independence needs justification. Two services can fail together because they share a power source. Two classifier errors can be correlated because their inputs come from the same user. Multiplying marginal probabilities is justified only when the model asserts the appropriate independence, not simply because the events have different names. Conditional independence can hold after a shared cause is specified even when unconditional independence fails.

Probability quantifies uncertainty within a chosen model; it does not repair selection bias or identify causes automatically. An alert rate estimated on one population may transfer poorly to another with a different fault prevalence. Quantum measurement will also use probabilities, but those are obtained from amplitudes through an additional physical rule. The finite-event bookkeeping here remains useful while leaving the origin of those probabilities to its own subject.