A and B are two events such that odds odds against A are 2 : 1 odds in favour of `A cup B` are 3 : 1. If `x le P(B) le y`, then the ordered pair (x, y) is

Options

A. `(5//12, 3//4)`
B. `(2//3, 3//4)`
C. `(1//3, 3//4)`
D. none of these

Explanation

To solve the problem, we need to find the ordered pair (x, y) such that \( x \leq P(B) \leq y \) given the odds against event A and the odds in favor of the union of events A and B.

### Step-by-Step Solution:

1. **Understanding Odds Against A**:
- The odds against A are given as 2:1. This means that for every 2 chances that A does not occur, there is 1 chance that A does occur.
- Therefore, the probability of A, \( P(A) \), can be calculated as:
\[ P(A) = \frac{1}{2 + 1} = \frac{1}{3} \]

2. **Understanding Odds in Favor of A ∪ B**:
- The odds in favor of \( A \cup B \) are given as 3:1. This means for every 3 chances that \( A \cup B \) occurs, there is 1 chance that it does not occur.
- Thus, the probability of \( A \cup B \), \( P(A \cup B) \), can be calculated as:
\[ P(A \cup B) = \frac{3}{3 + 1} = \frac{3}{4} \]

3. **Using the Formula for Probability of Union**:
- We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- Substituting the known values:
\[ \frac{3}{4} = \frac{1}{3} + P(B) - P(A \cap B) \]

4. **Rearranging the Equation**:
- Rearranging gives us:
\[ P(B) - P(A \cap B) = \frac{3}{4} - \frac{1}{3} \]
- Finding a common denominator (which is 12):
\[ \frac{3}{4} = \frac{9}{12}, \quad \frac{1}{3} = \frac{4}{12} \]
- Therefore:
\[ P(B) - P(A \cap B) = \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \]

5. **Expressing P(B)**:
- We can express \( P(B) \) in terms of \( P(A \cap B) \):
\[ P(B) = \frac{5}{12} + P(A \cap B) \]

6. **Finding Bounds for P(B)**:
- Since \( P(A \cap B) \) cannot exceed \( P(A) \):
\[ P(A \cap B) \leq P(A) = \frac{1}{3} \]
- Thus, substituting this into the equation for \( P(B) \):
\[ P(B) \leq \frac{5}{12} + \frac{1}{3} \]
- Converting \( \frac{1}{3} \) to twelfths:
\[ \frac{1}{3} = \frac{4}{12} \]
- Therefore:
\[ P(B) \leq \frac{5}{12} + \frac{4}{12} = \frac{9}{12} = \frac{3}{4} \]

7. **Finding the Minimum Value of P(B)**:
- The minimum value of \( P(B) \) occurs when \( P(A \cap B) = 0 \):
\[ P(B) \geq \frac{5}{12} \]

8. **Final Ordered Pair**:
- Thus, we have:
\[ \frac{5}{12} \leq P(B) \leq \frac{3}{4} \]
- The ordered pair \( (x, y) \) is:
\[ \left( \frac{5}{12}, \frac{3}{4} \right) \]

### Final Answer:
The ordered pair \( (x, y) \) is \( \left( \frac{5}{12}, \frac{3}{4} \right) \).