Why P(A|B') not P(A)-P(A n B)?

  • Context: Undergrad 
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Discussion Overview

The discussion revolves around the interpretation of conditional probability, specifically why P(A|B') is not equivalent to P(A) - P(A ∩ B). Participants explore the implications of conditioning on the event B' and the differences in probability spaces involved.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states the equation P(A|B') = P(A ∩ B) / P(B') but questions why it isn't simply P(A) - P(A ∩ B).
  • Another participant clarifies that P(A) can be expressed as P(A ∩ B) + P(A ∩ B'), indicating that the proposed version only holds if P(B') = 1.
  • A different participant emphasizes that P(A | B') should not include B in the sample space since B has not occurred, thus challenging the initial reasoning.
  • Further, a participant elaborates on the distinction between the probability spaces for P(A|B') and P(A ∩ B'), noting that they refer to different sets and that the reasoning based on a Venn diagram does not account for this difference.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between P(A|B') and P(A) - P(A ∩ B), with no consensus reached on the correct interpretation or equivalence of these expressions.

Contextual Notes

Participants highlight the importance of understanding the underlying probability spaces when discussing conditional probabilities, indicating that assumptions about these spaces may affect the validity of the proposed equations.

CAH
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I know the equations that P(A|B') = P(AnB) / P(B')

But why isn't it P(A) - P(A n B)

See photo attachment

We know b didn't happen so isn't it just A minus the middle?
 

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P(A)=P(A ∩ B )+P(A ∩ B')
P(A|B')=P(A ∩ B')/P(B')

Your version will hold only of P(B')=1.
 
Specifically, P(A | B^{C}) := P(A) - P(A \cap B) would still include B in the sample space. Given that B hasn't happen, you don't want B in the sample space.
 
CAH said:
I know the equations that P(A|B') = P(AnB) / P(B')

But why isn't it P(A) - P(A n B)

In an expression for probability P(S), there are more things involved that the set S. The expression for a probability involves (perhaps implicitly) a particular "probability space". The expressions P(A|B') and P(A \cap B') both refer to the same set. However, they refer to different probability spaces. In the probability space for P(A|B') no events in A \cap B exist. In the probability space for A \cap B' , events in A \cap B may exist and may be assigned nonzero probabilities.

Your reasoning with the Venn diagram doesn't include the information about what sets are in the two different probability spaces.
 

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