If A and B are two events such that P(A) = 3/4 and P(B) = 5/8, then

Options

A. `P(A uu B) ge 3//4`
B. `P(A' nn B) le 1//4`
C. `1//8 le P(A nn B') le 3//8`
D. `3//8le P(A nn B) le 5//8`

Explanation

To solve the problem, we need to find the minimum and maximum values of the intersection of two events A and B, given their individual probabilities.

### Step-by-Step Solution:

1. **Identify Given Probabilities**:
- We have \( P(A) = \frac{3}{4} \)
- We have \( P(B) = \frac{5}{8} \)

2. **Use the Formula for the Union of Two Events**:
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

3. **Determine Maximum Value of \( P(A \cap B) \)**:
The maximum value of \( P(A \cap B) \) can be determined by the smaller of the two probabilities:
\[ P(A \cap B) \leq \min(P(A), P(B)) \]
Since \( P(A) = \frac{3}{4} \) and \( P(B) = \frac{5}{8} \), we find:
\[ P(A \cap B) \leq \frac{5}{8} \]

4. **Determine Minimum Value of \( P(A \cap B) \)**:
To find the minimum value of \( P(A \cap B) \), we can use the inequality:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]
Substituting the values:
\[ P(A \cap B) \geq \frac{3}{4} + \frac{5}{8} - 1 \]
Converting \( \frac{3}{4} \) to eighths gives \( \frac{6}{8} \):
\[ P(A \cap B) \geq \frac{6}{8} + \frac{5}{8} - \frac{8}{8} = \frac{3}{8} \]

5. **Summarize the Results**:
From the calculations:
- The minimum value of \( P(A \cap B) \) is \( \frac{3}{8} \).
- The maximum value of \( P(A \cap B) \) is \( \frac{5}{8} \).

### Final Result:
Thus, we conclude that:
\[ \frac{3}{8} \leq P(A \cap B) \leq \frac{5}{8} \]