For events $A$ and $B$ with $P(B)>0$, conditional probability is $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$$ Conditioning restricts attention to cases where $B$ occurred, then asks what fraction also satisfy $A$. It does not imply that $B$ caused $A$.
Using $P(A\cap B)=P(B\mid A)P(A)$ gives the update $$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}.$$ Here $P(A)$ is the prior, $P(B\mid A)$ is likelihood of evidence under $A$, and $P(B)$ normalizes the result. For mutually exclusive hypotheses $H_i$, evidence is $P(B)=\sum_iP(B\mid H_i)P(H_i)$.
Posterior odds separate prior belief from evidence: $$\frac{P(A\mid B)}{P(\neg A\mid B)}=\frac{P(A)}{P(\neg A)}\cdot\frac{P(B\mid A)}{P(B\mid\neg A)}.$$ The second factor is an evidence ratio. In log space, updating becomes addition, which is numerically stable when probabilities are very small.
Base rates matter. A test can have high detection probability and low false-positive probability yet produce many false alarms when the condition itself is rare. Always combine likelihoods with the prior frequency. Reporting only sensitivity-like quantities without a base rate does not determine posterior probability.
Variables $X$ and $Y$ are conditionally independent given $Z$ when $p(x,y\mid z)=p(x\mid z)p(y\mid z)$. They may still be dependent before conditioning because both reflect $Z$. Conditional independence enables factorization of large joint distributions, but assuming it incorrectly can create overconfident posteriors.
In learned systems, estimated likelihoods and priors carry model error. Check probability calibration, evaluate under changed base rates, and avoid treating a posterior as certainty. State the hypothesis set explicitly: probabilities can be normalized perfectly while excluding the true explanation from every modeled alternative.