For two events A and B `P(A)=3/4` and `P(B)=5/8`. The false statement is:

Options

A. `P(AuuB)ge3/4`
B. `1/8leP(AnnB')le3/8`
C. `3/8leP(AnnB)le5/8`
D. `P(AuuB)le1/2`

Explanation

To determine which statement is false among the given options regarding the probabilities of events A and B, we can follow these steps:

### Given:
- \( P(A) = \frac{3}{4} \)
- \( P(B) = \frac{5}{8} \)

### Step 1: Calculate \( P(B') \)
First, we need to find the probability of the complement of event B, denoted as \( P(B') \):
\[ P(B') = 1 - P(B) = 1 - \frac{5}{8} = \frac{3}{8} \]

### Step 2: Analyze Option 1
**Option 1:** \( P(A \cup B) \geq \frac{3}{4} \)

Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since \( P(A) = \frac{3}{4} \) and \( P(B) = \frac{5}{8} \), we can say:
\[ P(A \cup B) \geq P(A) \quad \text{(since \( P(A \cap B) \) cannot be negative)} \]
Thus, \( P(A \cup B) \geq \frac{3}{4} \) is **true**.

### Step 3: Analyze Option 2
**Option 2:** \( P(A \cap B') \geq \frac{1}{8} \) and \( P(A \cap B') \leq \frac{3}{8} \)

Using the relationship:
\[ P(A \cap B') = P(A) - P(A \cap B) \]
To find \( P(A \cap B) \), we use:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]
From our previous analysis, we know \( P(A \cup B) \geq \frac{3}{4} \). Hence:
\[ P(A \cap B) \leq P(A) + P(B) - \frac{3}{4} \]
Calculating \( P(A) + P(B) \):
\[ P(A) + P(B) = \frac{3}{4} + \frac{5}{8} = \frac{6}{8} + \frac{5}{8} = \frac{11}{8} \]
Thus:
\[ P(A \cap B) \leq \frac{11}{8} - \frac{3}{4} = \frac{11}{8} - \frac{6}{8} = \frac{5}{8} \]
Now substituting back:
\[ P(A \cap B') = P(A) - P(A \cap B) \geq \frac{3}{4} - \frac{5}{8} = \frac{6}{8} - \frac{5}{8} = \frac{1}{8} \]
And:
\[ P(A \cap B') \leq \frac{3}{8} \]
Thus, this option is also **true**.

### Step 4: Analyze Option 3
**Option 3:** \( P(A \cap B) \geq \frac{3}{8} \) and \( P(A \cap B) \leq \frac{5}{8} \)

From our earlier calculations:
\[ P(A \cap B) \leq \frac{5}{8} \quad \text{(true)} \]
Now, to check if \( P(A \cap B) \geq \frac{3}{8} \):
Using the previous inequality:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \geq \frac{3}{4} + \frac{5}{8} - 1 = \frac{11}{8} - 1 = \frac{3}{8} \]
So, this option is also **true**.

### Step 5: Analyze Option 4
**Option 4:** \( P(A \cup B) \leq \frac{1}{2} \)

From our earlier analysis, we established that:
\[ P(A \cup B) \geq \frac{3}{4} \]
Thus, this statement is **false**.

### Conclusion
The false statement is:
\[ P(A \cup B) \leq \frac{1}{2} \]