Transitive Conditional Probability Constraints Is there a nice rule describing how the values of $P(C|B)$ and $P(B|A)$ jointly constrain $P(C|A)$? In particular, if I know that both $P(C|B)$ and $P(B|A)$ are above some threshold value $t$, what does that tell us about $P(C|A)$?

You cannot conclude anything about $P(C|A)$, it can still be $0$. For example, if I am rolling a $6$ sided die and the events $A,B,C$ are

* $A$... I roll an odd number
* $B$... I roll a number between $1$ and $6$
* $C$... I roll an even number



Then $P(C|B)=\frac12, P(B|A)=1$ (they are above the threshold $t=\frac12$), however $P(C|A)=0$.

There is also no upper bound, since you can replace $A$ above with "rolling an even number",a nd $P(C|A($ is then $1$.