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As shown in the image below, how does A turn into B? What is the intuitive logic behind the disappearance of sigma y? Thanks!!

enter image description here

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    This is e$x$actly correct. Formally this is called the Law of Total Probability.2012-12-02

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The event $\{X=x\}$ is the union of the mutually exclusive events $\{X=x,\ Y=y\}$ for all $y$.