Multiplication of conditional probability

Question

I was looking up on Bayesian networks and came across this video. In which it states:

And I was wondering, how did he get $P(B|A)*P(C|B) = P(B,C|A)$?

Answer 1

And I was wondering, how did he get $P(B|A)\cdot P(C|B)=P(B,C|A)$?

By reading the partial diagram $\require{enclose}{\to}\enclose{box}{\enclose{circle}{A\,}\to\enclose{circle}{B\,}\to\enclose{circle}{C\,}}{\to}$ which would factorise as $\mathsf P(A,B,C)=\mathsf P(A)\cdot\mathsf P(B\mid A)\cdot\mathsf P(C\mid B)$ .