Probability – Understanding Markov Chain and Conditional Probabilities

Question

If I define a Markov chain,
$$
A\to B\to C
\\P(A,B,C)=P(A)P(B|A)P(C|B)
$$
Can I derive an expression for $P(C|A)$?

I feel like this should be $P(C|B)P(B|A)$, but I am not seeing how to prove it.

Answer 1

Your direction is correct, but you need to sum that expression over all possible $b$:

$$P(C=c|A=a)=\sum_b P(B=b|A=a)P(C=c|B=b)$$

It should cover all the paths from $A=a$ to $C=c$.