= Sharp bounds from binary conditional probabilities
{title2=$P(B)=qr+(1-q)s$}
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 $[0,1]$,
$$
qr\leq P(B)\leq qr+1-q.
$$
With $r,s$ fixed and $q$ free in $[0,1]$,
$$
\min(r,s)\leq P(B)\leq\max(r,s).
$$
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.
Back to article page
