 Sharp bounds from binary conditional probabilities

Write $q=P(A)$, $r=P(B\mid A)$ and $s=P(B\mid A^c)$. The law of total probability gives the displayed convex combination. With $q,r$ fixed and $s$ free in $[0,1]$,With $r,s$ fixed and $q$ free in $[0,1]$,The joint cell masses $qr,q(1-r),(1-q)s,(1-q)(1-s)$ are nonnegative and sum to one, proving sharpness in the joint-model interpretation. For ordinary conditional probabilities, a conditioning event must have positive mass. If both $r$ and $s$ are specified by ratios, then $0<q<1$: for $r\ne s$ the last bounds are strict and their endpoints are the infimum and supremum. If $r=s$, the marginal equals that common value for every $q$. Endpoint mixtures remain meaningful if the unused zero-mass conditional is treated as a prescribed conditional model rather than a defined ratio.