1
$\begingroup$

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!

  • 0
    @okay-a$t$-math: I didn't pu$s$h the calculation through.2011-11-08

1 Answers 1

2

It’s false as stated. Suppose that the space is the sample space for two tosses of a fair coin,

  • $B_1$ is the event that the first toss is heads,
  • $B_2$ is the event that the first toss is tails,
  • $A$ is the event that both tosses are heads, and
  • $C$ is the event that at least one toss is tails (i.e, the complementary event to $A$).

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.

  • 0
    We would be happy to take a look at your actual question i$f$ you are still bothered about it.2011-11-08