Understanding the Difference Between P (A, B) and P (B, A)

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    Bayes theorem Theorem
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Discussion Overview

The discussion revolves around the interpretation of the notation P(A, B) versus P(B, A) in the context of probability theory, particularly in relation to Bayes' Theorem. Participants explore the implications of these notations and their equivalence in representing joint probabilities.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that P(A, B) and P(B, A) represent the same joint probability, as the events "A and B" and "B and A" are equivalent.
  • Others propose that P(A, B) is a functional notation that should be defined, suggesting that the order of variables typically does not matter unless specified.
  • A participant reiterates the definitions of P(A, B) and P(B, A) in terms of conditional probabilities, questioning the reasoning behind their equivalence.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of P(A, B) and P(B, A). While some agree on their equivalence in representing joint probabilities, others emphasize the need for careful definition and context, indicating that the discussion remains unresolved.

Contextual Notes

There are limitations regarding the assumptions made about the notation and its definitions, as well as the context in which these probabilities are applied. The discussion does not resolve these nuances.

spaghetti3451
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In proving Bayes' Theorem,

we use the following two statements.

P (A, B) = P (A|B) P (B)
P (B, A) = P (B|A) P (A).

I am wondering what's the difference between P (A, B) and P (B, A).

Any takers?
 
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There is no difference if by P(A,B) means the probability of the event "A and B". The event "A and B" is the same as the event "B and A".
 
Thanks!
 
failexam said:
In proving Bayes' Theorem,

we use the following two statements.

P (A, B) = P (A|B) P (B)
P (B, A) = P (B|A) P (A).

I am wondering what's the difference between P (A, B) and P (B, A).

Any takers?

P(A,B) is functional notation which is to be defined such as in [tex]f(x,y)= 6x + y^2[/tex] for example. The order of variables in the argument doesn't usually matter unless specifically stated.

You've defined it in terms of probabilities two ways which can be written:

[tex]P(A\cap B)[/tex] and [tex]P(B \cap A)[/tex]

They are the same but not because P(A,B) means P(A^B). P(A,B) is simply a function which is to be defined.
 
Last edited:

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