Simple probability question: Suppose P(B|A)=1. Does that mean that P(A|B)=1?

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Discussion Overview

The discussion revolves around the implications of conditional probabilities, specifically whether P(A|B) can be inferred from P(B|A) = 1. Participants explore concepts related to probability theory, causation, and the relationships between events A and B.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants question whether P(A|B) = 1 necessarily follows from P(B|A) = 1, suggesting that A could be a subset of B.
  • Others propose that if P(A) = P(B), then it might hold true, but this is not universally accepted.
  • Bayes' Theorem is mentioned, with some arguing that if P(A) = P(B), then P(A|B) could also equal 1.
  • There is a discussion about the nature of causality, with some asserting that A cannot be both the cause and effect of B.
  • Participants express confusion about the definitions of causality and how they relate to conditional probabilities.
  • Some argue that the concept of "possible" outcomes in probability theory is not formally defined, leading to philosophical discussions about the nature of events.
  • There is a suggestion that the definition of causality may vary, and that retrocausality is generally ruled out under standard definitions.
  • One participant emphasizes that conditional probability should be viewed as an updated probability rather than implying causation.
  • Another participant highlights the ambiguity in defining "cause" within the context of probability theory.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether P(A|B) = 1 follows from P(B|A) = 1. Multiple competing views on causality and the implications of conditional probabilities remain unresolved.

Contextual Notes

Discussions include limitations in definitions of causality, the implications of events with zero probability, and the philosophical aspects of probability theory that are not strictly defined within the formal mathematical framework.

  • #31
entropy1 said:
But actually, if ##A \rightarrow B## AND ##C \rightarrow NOT(B)##, then I wonder if C=True results in A=NOT True (or, of course, A=True in C=NOT True).
We can do a short proof here:

To be proven: ##C \rightarrow \neg A##

##1: A \rightarrow B## assumption 1
##2: C \rightarrow \neg B## assumption 2
##3: C## assumption 3
##4: \neg B## modus ponens 2,3
##5: \neg A ## modus tollens 1,4
##6: C \rightarrow \neg A## hypothethical syllogism (1,2),3,(4),5

##-## and there you have it ##\dots##
 
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  • #32
entropy1 said:
No, ##A \rightarrow B## is equivalent to ##NOT(B) \rightarrow NOT(A)##.

But actually, if ##A \rightarrow B## AND ##C \rightarrow NOT(B)##, then I wonder if C=True results in A=NOT True (or, of course, A=True in C=NOT True).
I always wondered why probability and logic are in the same forum. Now I see why. :smile:
 
  • #33
sysprog said:
We can do a short proof here:

To be proven: ##C \rightarrow \neg A##

##1: A \rightarrow B## assumption 1
##2: C \rightarrow \neg B## assumption 2
##3: C## assumption 3
##4: \neg B## modus ponens 2,3
##5: \neg A ## modus tollens 1,4
##6: C \rightarrow \neg A## hypothethical syllogism (1,2),3,(4),5

##-## and there you have it ##\dots##
So does that mean that one of (1) or (2) gets "reversed"? Can we then speak of retrocausality? (reversed causality?)
 
  • #34
entropy1 said:
Can we then speak of retrocausality? (reversed causality?)
Please do not ask this question again without a clear and exact definition of retrocausality. Preferably one from the professional literature.

To all other participants: please do not respond to this question without such a definition.
 
  • #35
At this point we will go ahead and close this thread. I strongly recommend studying the existing literature in this topic, perhaps including the time symmetric formulation of quantum mechanics. It is best to use definitions from the literature as they are more likely to have addressed some of the basic issues mentioned so far.

For any future threads on this topic please start with a professional scientific reference that can serve as the basis of discussion
 

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