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 analyze the probabilities of events A and B given that \( P(A) = \frac{3}{4} \) and \( P(B) = \frac{5}{8} \). We will derive some relationships between these probabilities.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We are given two events A and B with their respective probabilities. We need to find the relationship between these probabilities, particularly focusing on \( P(A \cup B) \) and \( P(A \cap B) \).

2. **Using the Formula for Union of Two Events**:
The probability of the union of two events can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

3. **Substituting Known Values**:
We substitute the known values into the formula:
\[ P(A \cup B) = \frac{3}{4} + \frac{5}{8} - P(A \cap B) \]

4. **Finding a Common Denominator**:
To perform the addition, we need a common denominator. The least common multiple of 4 and 8 is 8.
- Convert \( \frac{3}{4} \) to eighths:
\[ \frac{3}{4} = \frac{6}{8} \]
- Now substitute:
\[ P(A \cup B) = \frac{6}{8} + \frac{5}{8} - P(A \cap B) = \frac{11}{8} - P(A \cap B) \]

5. **Understanding the Range of Probabilities**:
Since probabilities cannot exceed 1, we have:
\[ P(A \cup B) \leq 1 \]
Therefore:
\[ \frac{11}{8} - P(A \cap B) \leq 1 \]

6. **Solving for \( P(A \cap B) \)**:
Rearranging gives us:
\[ P(A \cap B) \geq \frac{11}{8} - 1 = \frac{3}{8} \]

7. **Finding the Upper Bound**:
Similarly, since \( P(A \cap B) \) cannot exceed either \( P(A) \) or \( P(B) \), we also have:
\[ P(A \cap B) \leq P(A) \quad \text{and} \quad P(A \cap B) \leq P(B) \]
Thus:
\[ P(A \cap B) \leq \frac{3}{4} \quad \text{and} \quad P(A \cap B) \leq \frac{5}{8} \]

8. **Combining the Results**:
From the inequalities, we have:
\[ \frac{3}{8} \leq P(A \cap B) \leq \frac{5}{8} \]

### Conclusion:
The range of \( P(A \cap B) \) is:
\[ \frac{3}{8} \leq P(A \cap B) \leq \frac{5}{8} \]