Understanding the Expansion of P(A)

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Discussion Overview

The discussion revolves around the expansion of the probability P(A) in the context of conditional probabilities and the assumptions underlying such expansions. Participants explore the implications of using subsets and complements in probability theory, focusing on the conditions necessary for these expansions to hold true.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the validity of expanding P(A) as P(A/B).P(B) + P(A/B').P(B'), suggesting it assumes A is a subset of B.
  • Another participant argues that the expansion reflects the total probability of A occurring, considering both scenarios where B occurs and where it does not.
  • A participant cites their textbook, stating that P(A) can be expressed as a sum over mutually exclusive events B1, B2, ..., Bn, questioning whether this implies that the union of these events must encompass A.
  • Further clarification is provided that the events B1, B2, ..., Bn should be mutually disjoint and cover the entire sample space, indicating that A must be a subset of this space for the probabilities to be meaningful.
  • One participant acknowledges the clarification, indicating that it resolves their confusion about the assumptions involved.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions required for the expansion of P(A), with some agreeing on the necessity of A being a subset of the universe of outcomes, while others focus on the broader implications of the total probability theorem.

Contextual Notes

The discussion highlights potential limitations in the assumptions regarding the relationships between events A and B, particularly concerning the necessity for B1, B2, ..., Bn to cover the entire sample space and the implications of A's relationship to these events.

Avichal
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P(B/A) = P(A/B).P(B) / P(A)

Later we expand P(A) as P(A/B).P(B) + P(A/B).P(B) ... B is complement of B

I don't understand how we can expand P(A) like that. Doesn't that assume that A ℂ B?
 
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Think about what it is saying. The probability that A happens is the probability that A happens given that B happens plus the probability that A happens given that B doesn't happen. Both cases are needed to cover all possibilities.
 
Well basically what my book says is that : -
P(A) = P(A/B1).P(B1) + P(A/B2).P(B2) + ... + P(A/Bn).P(Bn)

Doesn't this assume that B1 U B2 ... U Bn is a super-set of A?
 
Avichal said:
Doesn't this assume that B1 U B2 ... U Bn is a super-set of A?
Of course.

Your text should have specified that B1, B2, ···, Bn are a set of mutually disjoint subsets of the universe U of possible outcomes and that B1 ∪ B2 ∪ ··· ∪ Bn=U. The set A must be a subset of this universe of outcomes U; otherwise it doesn't even make sense to talk about P(B1|A).
 
D H said:
Of course.

Your text should have specified that B1, B2, ···, Bn are a set of mutually disjoint subsets of the universe U of possible outcomes and that B1 ∪ B2 ∪ ··· ∪ Bn=U. The set A must be a subset of this universe of outcomes U; otherwise it doesn't even make sense to talk about P(B1|A).

It didn't. Anyways, thank you. This clears my doubt.
 

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