Probability Proof for Events A, B, and C: Homework Help and Explanation

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    Probability Proof
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Homework Help Overview

The discussion revolves around a probability proof involving three events, A, B, and C. Participants are tasked with demonstrating two statements related to the probabilities of these events and their complements, as well as determining a specific probability given certain conditions.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants express uncertainty about how to start the proof and question the relevance of event C in the context of the first statement. There are attempts to manipulate the equations and consider complements, with some participants suggesting substitutions for the probabilities involved.

Discussion Status

The discussion is ongoing, with participants providing hints and clarifications. Some guidance has been offered regarding the manipulation of the probability expressions, and there is an acknowledgment of confusion regarding the role of event C in the proof.

Contextual Notes

Participants are navigating the complexities of probability axioms and the implications of complements in their proofs. There is a noted assumption that event C may need to be considered, although it is clarified that it is not relevant for the first part of the problem.

a little lost
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Homework Statement


Let A, B and C be any three events. Show that

i) P(A) = P(B) if and only if P(A U Bc) = P(Ac U B)

ii) Given P(A) = 0.5 and P(A U (Bc ∩ Cc)c = 0.8
determine P(Ac ∩ (B U C))

Homework Equations


the probability axioms?

The Attempt at a Solution



i) not sure where or how to start

ii) P(A U (Bc∩Cc)c) = P(A U B U C) = 0.8

then, P(Ac ∩ (B U C)) = P(B U C) - P(A) = 0.8 - 0.5 = 0.3

I think I'm wrong... though I'm not sure...
 
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hi a little lost! :smile:

for (i), try taking the complement of the second equation :wink:

(and (ii) looks fine :smile:)
 
@tiny_tim: oops, i just realized i wrote the iff "P(A U Bc) = P(Ac U B)" wrong
it should have been the complement as you said ^^"
-have been staring at this question for the past few days wondering what i should do next...

so, would i then substitute
P(A ∩ Bc)= P(A) - P(A ∩ B)
and likewise for P(Ac ∩ B) ?

if so, how does event C appear in/affect the proof?
 
hi a little lost! :smile:
a little lost said:
so, would i then substitute
P(A ∩ Bc)= P(A) - P(A ∩ B)
and likewise for P(Ac ∩ B) ?

yes

A = A ∩ (the whole space) = A ∩ (B U Bc) = (A ∩ B) U (A ∩ Bc) :wink:
if so, how does event C appear in/affect the proof?

i'm confused …

are we talking about question i) or ii) ? :confused:
 
tiny-tim said:
are we talking about question i) or ii) ? :confused:

i mean i) i just assumed since it was stated as an event it may have to appear in the proof for part i)
 
then no, it's not mentioned in i), so you needn't bother with it until ii) :smile:
 
ok thank-you very much for the help :D
 

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