Q: If A and B are two events, prove that $\displaystyle P(A\cap\\B)\geq\\1-P(A^{c})-P(B^{c})$.
I am having a rough start with this one. Any direction would be much appreciated.
Replace on the right, $\displaystyle P(A')=1-P(A)$ and the similar equation with B.
Hence you need to show that
$\displaystyle P(A)+P(B)-1\le P(AB)$
On the left use that $\displaystyle P(A)+P(B)=P(A\cup B)+P(AB)$
So, you need to show that
$\displaystyle P(A\cup B)+ P(AB)-1\le P(AB)$
and that reduces to
$\displaystyle P(A\cup B)\le 1$ WHICH is where you should start this proof.
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$\displaystyle P(A\cup B)\le 1$
$\displaystyle P(A)+P(B)-P(AB)\le 1$
$\displaystyle \bigl(1-P(A')\bigr)+\bigl(1-P(B')\bigr)-P(AB)\le 1$
$\displaystyle 1-P(A')+1-P(B')-P(AB)\le 1$
$\displaystyle 1-P(A')-P(B')\le P(AB)$