I think the statement $P(A|C) = \sum_{i} P(A|B_i)P(B_i|C)$ is true (where the events $B_i$ form a partition of the whole space). Is it, and why?
Thanks!
I think the statement $P(A|C) = \sum_{i} P(A|B_i)P(B_i|C)$ is true (where the events $B_i$ form a partition of the whole space). Is it, and why?
Thanks!
It’s false as stated. Suppose that the space is the sample space for two tosses of a fair coin,
Clearly $\mathbb{P}(A|C)=0$, but $\mathbb{P}(A|B_1)\mathbb{P}(B_1|C)+\mathbb{P}(A|B_2)\mathbb{P}(B_2|C)=\frac12\cdot\frac13+0\cdot\frac23=\frac16.$
Added: It never hurts to check some simple cases of a conjecture, if there are any. Those that are somehow atypical (like making $A$ and $C$ complementary events) are often especially useful.