on our actions we took and sensory readings we get well this is exactly the law of total probability we're applying here the conditional distribution for X prime next day given the action U is well we have a distribution for current state that's what we assume we assume that we know the solution for current state will then multiply in X prime given U and X to get to joint over X Prime and X once if they joined over X Prime and X we actually just want X Prime so we sum out or integrate out X to just get the thing