The law of total probability with extra conditioning

In summary, the conversation discusses the proof of the law of total probability with extra conditioning in the context of studying probability. It is mentioned that the proof requires careful consideration of the exact statement and that the ##A_i## need to be a partitioning. It is also clarified that when all probabilities are conditional on ##E##, it is essentially just another probability with ##E## as the universe and does not need to be included in the notation.
  • #1
red65
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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 :
1672964293581.png

thanks!
 
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  • #2
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?
 
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  • #3
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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