Proving two events are independent given that one has a probability of 1

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Discussion Overview

The discussion revolves around proving the independence of two events, A and B, under the condition that the probability of event A is 1. Participants explore different approaches to demonstrate this independence, focusing on the mathematical relationships between the probabilities involved.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant asserts that if P(A) = 1, then P(A n B) = P(B/A), leading to the need to show P(B/A) = P(B) for independence.
  • Another participant suggests an alternative approach using the equation P(A n B) = P(B)P(A/B).
  • A later reply indicates a challenge in proving that P(A/B) = 1 without assuming independence.
  • One participant reports successfully solving the problem using the complement method, though details of this method are not provided.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to prove independence when P(A) = 1, and multiple approaches are discussed without resolution.

Contextual Notes

Some assumptions about the relationships between the probabilities are not fully explored, and the discussion includes unresolved steps in the mathematical reasoning.

LCBlazer07
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Show that an event A is independent of every event B if P(A)=0 or P(A)=1.


**I was able to prove the first part of this problem that is that the events are independent when P(A)=0. However I am stuck on the part where P(A)=1.


I have this so far:

P(A n B) = P(A)P(B/A)
= (1)P(B/A)
= P(B/A)

If the events are independent then P(A n B) = P(A)P(B) = (PB)

So basically i have to show P(B/A) = P(B)...but I do cannot find a way to do this.
 
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Try it the other way:
P(A n B) = P(B)P(A/B)
 
Ok, so i started using that suggestion and got the following...

P( A n B) = P(B)P(A/B)
then I was going to say that P(A n B) = P(B), but I how can I prove that P(A/B)=1 without assuming that A and B are independent.
 
thanks for the suggestion, but I was able to solve this problem by using the complement.
 

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