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I think that P(X|Z)=P(X|Y)P(Y|Z) is true, and doing some calculations in two different ways, it appears to be correct. However, I'm not sure and can't seem to prove it. Is it true, and why?

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The statement is false. Using the definition of conditional probability, we have

$$P(X | Z) = \dfrac{P(X \cap Z)}{P(Z)}$$

and

$$\begin{aligned} P(X | Y)P(Y | Z) &= \dfrac{P(X \cap Y)}{P(Y)} \cdot \dfrac{P(Y \cap Z)}{P(Z)} \end{aligned}$$

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This is in general false. Toss a fair coin twice. Let $X$ be the event head on first, $Y$ the event head on second, $Z$ the event head on both.

Then $\Pr(X|Z)=1$, but $\Pr(X|Y)\Pr(Y|Z)\ne 1$.

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