If $A$ and $B$ are two events such that $P(A \cup B) = \frac{3}{4}$, $P(A \cap B) = \frac{1}{4}$, and $P(\overline{A}) = \frac{1}{3}$, then what is the relationship between events $A$ and $B$?

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Accepted Answer

**Solution:**

1. Given:
   - $P(A \cup B) = \frac{3}{4}$
   - $P(A \cap B) = \frac{1}{4}$
   - $P(\overline{A}) = \frac{1}{3}$

2. First, find $P(A)$ using the complement rule:
   $$P(A) = 1 - P(\overline{A}) = 1 - \frac{1}{3} = \frac{2}{3}$$

3. Use the formula for the union of two events:
   $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Substitute the known values:
   $$\frac{3}{4} = \frac{2}{3} + P(B) - \frac{1}{4}$$

4. Solve for $P(B)$:
   $$P(B) = \frac{3}{4} - \frac{2}{3} + \frac{1}{4} = \left(\frac{3}{4} + \frac{1}{4}ight) - \frac{2}{3} = 1 - \frac{2}{3} = \frac{1}{3}$$

5. Check if $A$ and $B$ are independent by verifying if:
   $$P(A \cap B) = P(A) 	imes P(B)$$

Calculate $P(A) 	imes P(B)$:
   $$\frac{2}{3} 	imes \frac{1}{3} = \frac{2}{9}$$

Since $P(A \cap B) = \frac{1}{4} = 0.25$ and $\frac{2}{9} \approx 0.222$, they are not equal.

6. Check if $A$ and $B$ are mutually exclusive:
   - Mutually exclusive events satisfy $P(A \cap B) = 0$.
   - Here, $P(A \cap B) = \frac{1}{4} 
eq 0$, so they are not mutually exclusive.

7. Since the events are neither independent nor mutually exclusive, they must be dependent.

**Answer:** Dependent

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