Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 2 / 17 / c / ii

Write $q=P(B\mid A)$, $s=P(B\mid A^c)$ and $p=P(A)$. The law of total probability gives $P(B)=pq+(1-p)s$, a convex combination of the two fixed conditional values. HenceThese are the closed sharp bounds in the usual mixture-model interpretation, obtained by allowing $p$ to range from zero to one. The joint masses $pq,p(1-q),(1-p)s,(1-p)(1-s)$ realize every such mixture.

There is an endpoint qualification if both conditionals must literally be defined by ratios: then $0<p<1$. If $q\ne s$, the marginal lies strictly between them, so the stated endpoints are an infimum and supremum rather than achieved extrema. If $q=s$, every allowed mixture has exactly that common value. This is the second case of sharp bounds from binary conditional probabilities.