B The law of total probability with extra conditioning

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The discussion centers on the law of total probability with extra conditioning, where the user seeks a proof. Key points highlight the necessity for the events to partition the sample space and the clarification that conditioning on a specific event can simplify the notation. Participants emphasize that the proof hinges on understanding these conditions. The conversation underscores the importance of precise definitions in probability theorems. Overall, the thread aims to clarify the theorem's proof requirements and implications.
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the proof of a theorem
Hello, I am studying probability and came across this theorem, it's the law of total probability with extra conditioning, I tried to work out a proof but couldn't ,does anyone know the proof for this :
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thanks!
 
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1) If you are looking for a proof, you should be very careful about the exact statement. There is more to that statement, right? Don't the ##A_i## need to be a partitioning?
2) If all probabilities are conditional on ##E##, isn't that just another probability where ##E## is the universe and does not need to be included in the notation?
 
red65 said:
TL;DR Summary: the proof of a theorem

Hello, I am studying probability and came across this theorem, it's the law of total probability with extra conditioning, I tried to work out a proof but couldn't ,does anyone know the proof for this :
View attachment 319864
thanks!
That's just the usual equation with a restriction to ##E## as the universal set or sample space. Given the proviso, as above, that the ##A_i## (when restricted to ##E##) partition ##E##.
 
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