# Problem with a proof in probability

1. Sep 18, 2008

### DJ_JK

1. The problem statement, all variables and given/known data
I have to prove that for any two events A and B
P(A and B|A) $$\geq$$ P(A and B| A or B)

2. Relevant equations
P(A and B) = P(A) . P(B|A)
P(A|B) = (P(A) . P(B|A))/P(B)

3. The attempt at a solution
I tried to simplify the left side with this reasoning
P(A and B|A) = P(A) . P((B|A)|A)
= P(A) . P(B|A)
= P(A and B)

My reasoning for going from step 1 to two is that condition A is already fulfilled, and asking for it a second time is needless. My friend however, disagrees with this.
I am having problems with simplifying the right side because I don't know if there is a system of priorities in probability mathematics (e.g. "condition" has a priority over "and", "or" over "condition", ...
Any help in the right direction would be greatly welcomed.

2. Sep 18, 2008

Remember that for any two events $$\mathcal{C}, \mathcal{D}$$ you have
$$\Pr(\mathcal{C} | \mathcal{D}) = \frac{\Pr(\mathcal{C} \cap \mathcal{D})}{\Pr{\mathcal{D}}}$$