For two events A and B, let P(A) = `3/5`, P(B) = `2/3`, then which of the following is correct?

Options

A. `P(A cap barB) le (1)/(3)`
B. `P(A cup B) ge (2)/(3)`
C. `(4)/(15) le P(A cap B) le (3)/(5)`
D. `(1)/(10) le P(barA//B) le (3)/(5)`

Answer

To solve the problem, we need to analyze the probabilities of events A and B given that \( P(A) = \frac{3}{5} \) and \( P(B) = \frac{2}{3} \). We will evaluate the four statements provided and determine which ones are correct.

### Step 1: Calculate \( P(A \cup B) \)

Using the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We know \( P(A) = \frac{3}{5} \) and \( P(B) = \frac{2}{3} \).

To find \( P(A \cup B) \), we first need to establish the range for \( P(A \cap B) \).

### Step 2: Determine the range for \( P(A \cap B) \)

1. **Maximum value of \( P(A \cap B) \)**:
\[ P(A \cap B) \leq \min(P(A), P(B)) = \min\left(\frac{3}{5}, \frac{2}{3}\right) = \frac{2}{3} \]

2. **Minimum value of \( P(A \cap B) \)**:
Using the formula:
\[ P(A \cap B) \geq P(A) + P(B) - 1 \]
Substituting the values:
\[ P(A \cap B) \geq \frac{3}{5} + \frac{2}{3} - 1 \]
To calculate this, we need a common denominator (15):
\[ P(A \cap B) \geq \frac{9}{15} + \frac{10}{15} - \frac{15}{15} = \frac{4}{15} \]

Thus, we have:
\[ \frac{4}{15} \leq P(A \cap B) \leq \frac{2}{3} \]

### Step 3: Calculate \( P(A \cup B) \)

Now, substituting the minimum value of \( P(A \cap B) \) into the union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \geq \frac{3}{5} + \frac{2}{3} - \frac{2}{3} \]
This simplifies to:
\[ P(A \cup B) \geq \frac{3}{5} = 0.6 \]
Since \( P(A \cup B) \) must be less than or equal to 1, we can conclude:
\[ P(A \cup B) \geq \frac{2}{3} \text{ (which is approximately 0.67)} \]

### Step 4: Evaluate the statements

1. **Statement 1**: \( P(A \cap B') \leq \frac{1}{3} \)
- This can be calculated as \( P(A) - P(A \cap B) \). Since \( P(A \cap B) \) has a minimum of \( \frac{4}{15} \), we find:
\[ P(A \cap B') = P(A) - P(A \cap B) \geq \frac{3}{5} - \frac{4}{15} = \frac{9}{15} - \frac{4}{15} = \frac{5}{15} = \frac{1}{3} \]
- Thus, this statement is **correct**.

2. **Statement 2**: \( P(A \cup B) \geq \frac{2}{3} \)
- As calculated, \( P(A \cup B) \geq \frac{2}{3} \) is **correct**.

3. **Statement 3**: \( \frac{4}{15} \leq P(A \cap B) \leq \frac{3}{5} \)
- This is true based on our earlier calculations, so this statement is **correct**.

4. **Statement 4**: \( P(A' | B) \) is between \( \frac{1}{10} \) and \( \frac{1}{5} \)
- This requires calculating \( P(A') \) given \( P(B) \). We find:
\[ P(A' | B) = \frac{P(A' \cap B)}{P(B)} = \frac{P(B) - P(A \cap B)}{P(B)} \]
- Given the range of \( P(A \cap B) \), we can establish that this statement is also **correct**.

### Conclusion

All statements are correct:
1. \( P(A \cap B') \leq \frac{1}{3} \) is correct.
2. \( P(A \cup B) \geq \frac{2}{3} \) is correct.
3. \( \frac{4}{15} \leq P(A \cap B) \leq \frac{3}{5} \) is correct.
4. \( P(A' | B) \) is between \( \frac{1}{10} \) and \( \frac{1}{5} \) is correct.