If A and B are two events such that P(A) = `(1)/(2)` and P(B) = `(2)/(3)`, then show that
(a) `P(Auu B) ge (2)/(3) (b) (1)/(6) le P(A nn B) le (1)/(2)`
(c ) `P(A nn barB) le (1)/(3) (d) (1)/(6) le P(barAnn B) le (1)/(2)`

Explanation

To solve the problem, we need to analyze the probabilities of events A and B given that \( P(A) = \frac{1}{2} \) and \( P(B) = \frac{2}{3} \). We will prove each part step by step.

### (a) Show that \( P(A \cup B) \geq \frac{2}{3} \)

1. **Use the formula for the union of two events:**
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
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