1. Nice Probability Inequalities

Hi all, I am new to this forum, but think It's a great idea! I have posted two questions, neither of which were unfortunately answered. But, it's ok, cos it made me figure out one of the questions on my own and plus people aren't obliged to answer your questions : ) here are three elementary inequalities that I'm trying to prove. For the 1st case I think we need induction, tips would be appreciated. The second one, I think I need an elementary identity to prove it, which I can't think of so I'm feeling sloooow and the third I haven't even attempted. Thanks for your help.

Prove:

a)

$\mathbb {P} [\bigcap _{i=1} ^ {n} A_i] \geq \mathbb {P}[A_1] + ... + \mathbb {P}[A_n] - (n-1)$

for any events $A_1, .... , A_n$

b)

${\frac {-1}_{4}} \leq \mathbb {P}[A\cap B]-\mathbb {P}[A]\mathbb {P}[B] \leq {\frac {1}_{4}}$

for any events A & B.

c)

$|\mathbb{P}[A\cap B]-\mathbb{P}[A\cap C]| \leq \mathbb{P}[(B-C)\cup (C-B)]$

for any events A, B and C.

2. Ok, so the first part just uses a straightforward induction like I thought; we just use P (A cup B) = P (A) + P (B) - P (A cap B) and it pops out. Still wondering about parts b) & c)....any thoughts yet?

3. Part C) done!

Part C is also not difficult. It's easy to prove that:

$|P(X) - P(Y)| \leq P(X \Delta Y)$ for any sets X & Y.

It's also clear that:

$(A \cap B) \Delta (A \cap C) \subset (B \Delta C)$ and the inequality follows.

Only part b) left.....come on guys.