# Probability Proof Help

1. Mar 15, 2013

### a little lost

1. The problem statement, all variables and given/known data
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))

2. Relevant equations
the probability axioms?

3. 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...

Last edited by a moderator: Mar 16, 2013
2. Mar 16, 2013

### tiny-tim

hi a little lost!

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

(and (ii) looks fine )

3. Mar 18, 2013

### a little lost

@tiny_tim: oops, i just realised 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?

4. Mar 18, 2013

### tiny-tim

hi a little lost!
yes

A = A ∩ (the whole space) = A ∩ (B U Bc) = (A ∩ B) U (A ∩ Bc)
i'm confused …

are we talking about question i) or ii) ?

5. Mar 18, 2013

### a little lost

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

6. Mar 18, 2013

### tiny-tim

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

7. Mar 18, 2013

### a little lost

ok thank-you very much for the help :D