How do I prove that the probability of the intersection of three events is equal to 1 minus the probability of the complement of each event?

Question

Having trouble with the following problem for my homework:

Prove $$P(A\cap B\cap C)\ge1 - P(A^c) - P(B^c) - P(C^c)$$

I've tried finding $P((A \cap B \cap C)^c)$ to compare, but I'm left with a bunch of terms that don't go away.

Any hints would be appreciated!

Answer 1

$P(A) = 1 - P(A^c)$

$P(A \cap B) = P(A) - P(A \cap B^c) \ge 1 - P(A^c) - P(B^c)$

$P(A \cap B \cap C) = \ldots$