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

[a little lost]

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 B^c) = P(A^c U B)

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

then, P(A^c ∩ (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...

 

[tiny-tim]

hi a little lost!

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

(and (ii) looks fine )

 

[a little lost]

@tiny_tim: oops, i just realized i wrote the iff "P(A U B^c) = P(A^c 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 ∩ B^c)= P(A) - P(A ∩ B)
and likewise for P(A^c ∩ B) ?

if so, how does event C appear in/affect the proof?

 

[tiny-tim]

hi a little lost!

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

yes

A = A ∩ (the whole space) = A ∩ (B U B^c) = (A ∩ B) U (A ∩ B^c)

if so, how does event C appear in/affect the proof?

i'm confused …

are we talking about question i) or ii) ?

 

[a little lost]

are we talking about question i) or ii) ?

i mean i) i just assumed since it was stated as an event it may have to appear in the proof for part i)

 

[tiny-tim]

then no, it's not mentioned in i), so you needn't bother with it until ii)

 

[a little lost]

ok thank-you very much for the help :D