If I know the conditional probability of one event P(A|B) and then I know the probability of the event B given another event C, i.e. P(B|C), can I easily calculate P(A|C) ?
1 Answer
$\begingroup$
$\endgroup$
No. Consider the following: Let $U$ be a standard uniform random variable (uniformly distributed on $[0, 1]$).
- $A=C=\{0\le U\le1/2\}$, $B=\{1/4\le U\le3/4\}$.
- $A=\{0\le U\le1/2\}$, $B=\{1/4\le U\le3/4\}$, $C=\{1/2\le U\le 1\}$.
In both cases, $P(A|B)=P(B|C)=1/2$, but $P(A|C)=1$ in the first case and $P(A|C)=0$ in the second case. In general, $P(A|B)$ and $P(B|C)$ are not sufficient for calculating $P(A|C)$.
