Question

If a and b are two events, prove that p(a ∩ b) ≥ 1 − p(a) − p(b).

Answer 1

To prove the inequality P(A ∩ B) ≥ 1 - P(A) - P(B), we can use the inclusion-exclusion principle and simplify the equation to obtain the desired result.

Step-by-step explanation:

To prove that P(A ∩ B) ≥ 1 - P(A) - P(B), we can use the following steps:

1. Start with the inequality P(A ∩ B) + P(A ∪ B) = P(A) + P(B) - P(A ∪ B). This is known as the inclusion-exclusion principle.
2. Since A and B are events, P(A ∪ B) represents the probability that either A or B (or both) occurs.
3. Simplify the equation to get P(A ∩ B) = P(A) + P(B) - P(A ∪ B) - P(A ∪ B).
4. Since P(A ∪ B) is greater than or equal to 0 and less than or equal to 1, we can replace it with a lower bound of 0 and an upper bound of 1. Therefore, P(A ∩ B) is greater than or equal to P(A) + P(B) - 1.
5. Finally, rearrange the inequality to get P(A ∩ B) ≥ 1 - P(A) - P(B).