We are asked to prove the inequality:

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

Proof:

1. Recall the complement rules:

   By definition, the complement of an event $A$ (denoted $A^c$) is the event that $A$ does not occur. Hence: $P(A^c) = 1 - P(A)$ Similarly, for event $B$: $P(B^c) = 1 - P(B)$

2. Use the fact that probabilities are bounded by 1:

   The probability of any event must satisfy $0 \leq P(E) \leq 1$ for any event $E$. Specifically: $P(A^c) \leq 1 \quad \text{and} \quad P(B^c) \leq 1$

3. Use the union bound (Boole's inequality):

   The probability of the union of two events $A^c$ and $B^c$ satisfies the following inequality: $P(A^c \cup B^c) \leq P(A^c) + P(B^c)$ By De Morgan's law, we know that: $A^c \cup B^c = (A \cap B)^c$ Therefore, the probability of the complement of $A \cap B$ can be written as: $P((A \cap B)^c) = P(A^c \cup B^c)$ Substituting this into the union bound inequality, we get: $P((A \cap B)^c) \leq P(A^c) + P(B^c)$

4. Relating the complement to the original event:

   The probability of the complement of $A \cap B$ is related to the probability of $A \cap B$ as follows: $P((A \cap B)^c) = 1 - P(A \cap B)$ Substituting this into the inequality derived from the union bound, we get: $1 - P(A \cap B) \leq P(A^c) + P(B^c)$

5. Rearranging the inequality:

   To isolate $P(A \cap B)$, subtract $P(A^c) + P(B^c)$ from both sides: $P(A \cap B) \geq 1 - P(A^c) - P(B^c)$

This completes the proof.

Conclusion:

We have shown that: $P(A \cap B) \geq 1 - P(A^c) - P(B^c)$

---

Would you like further details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does the inequality change if $A$ and $B$ are independent events?
2. What is the interpretation of $P(A^c \cup B^c)$ in a real-world example?
3. Can this inequality be strict, and under what conditions?
4. How does this inequality relate to conditional probabilities?
5. How would the inequality look for three events $A$, $B$, and $C$?

Tip: Understanding complement events is crucial for solving probability problems involving unions and intersections.