Odds in favour of an event A are 2 to 1 and odds in favour of `A cup B` are 3 to 1. Consistant with his information the smallest and largest values for the probability of event B are given by

Options

A. `(1)/(6) le P(B) le (1)/(3)`
B. `(1)/(3) le P(B) le (1)/(2)`
C. `(1)/(12) le P(B) le (3)/(4)`
D. none of these

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Odds
The odds in favor of event A are given as 2 to 1. This means:
- Probability of A, \( P(A) = \frac{2}{2+1} = \frac{2}{3} \)

### Step 2: Calculate Probability of A Union B
The odds in favor of \( A \cup B \) are given as 3 to 1. This means:
- Probability of \( A \cup B \), \( P(A \cup B) = \frac{3}{3+1} = \frac{3}{4} \)

### Step 3: Use the Formula for Probability of Union
We know the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ \frac{3}{4} = \frac{2}{3} + P(B) - P(A \cap B) \]

### Step 4: Express \( P(A \cap B) \)
Rearranging the equation gives:
\[ P(A \cap B) = P(A) + P(B) - P(A \cup B) \]
Substituting the known probabilities:
\[ P(A \cap B) = \frac{2}{3} + P(B) - \frac{3}{4} \]

### Step 5: Find a Common Denominator
To simplify, we need a common denominator for the fractions:
- The least common multiple of 3 and 4 is 12.
\[ P(A) = \frac{2}{3} = \frac{8}{12} \]
\[ P(A \cup B) = \frac{3}{4} = \frac{9}{12} \]
Now substituting these values:
\[ P(A \cap B) = \frac{8}{12} + P(B) - \frac{9}{12} \]
This simplifies to:
\[ P(A \cap B) = P(B) - \frac{1}{12} \]

### Step 6: Set Up Inequalities
Since \( P(A \cap B) \) must be greater than 0 and less than or equal to \( P(A) \):
1. \( P(B) - \frac{1}{12} > 0 \)
2. \( P(B) - \frac{1}{12} \leq \frac{2}{3} \)

From the first inequality:
\[ P(B) > \frac{1}{12} \]

From the second inequality:
\[ P(B) \leq \frac{2}{3} + \frac{1}{12} \]
Finding a common denominator for \( \frac{2}{3} \):
\[ \frac{2}{3} = \frac{8}{12} \]
Thus:
\[ P(B) \leq \frac{8}{12} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4} \]

### Step 7: Final Result
Combining the inequalities gives:
\[ \frac{1}{12} < P(B) \leq \frac{3}{4} \]

### Conclusion
The smallest value for the probability of event B is \( \frac{1}{12} \) and the largest value is \( \frac{3}{4} \).

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