For two events A and B, let `P(A) = (1)/(2) , P(A uuB) = (2)/(3)` and `P(A nnB) = (1)/(6)`. What is `P(bar(A) nn B)` equal to ?

Options

A. `(1)/(6)`
B. `(1)/(4)`
C. `(1)/(3)`
D. `(1)/(2)`

Explanation

To solve the problem, we need to find \( P(\bar{A} \cap B) \), given the following probabilities:

- \( P(A) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{2}{3} \)
- \( P(A \cap B) = \frac{1}{6} \)

### Step 1: Use the formula for the union of two events

We know the formula for the probability of the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 2: Substitute the known values into the formula

Substituting the known values into the formula:
\[ \frac{2}{3} = \frac{1}{2} + P(B) - \frac{1}{6} \]

### Step 3: Simplify the equation

First, convert all fractions to a common denominator. The least common multiple of 2, 3, and 6 is 6. Thus, we rewrite the equation:
\[ \frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{6} = \frac{1}{6} \]
Now, substituting these values:
\[ \frac{4}{6} = \frac{3}{6} + P(B) - \frac{1}{6} \]

### Step 4: Combine the fractions

Now, simplify the right side:
\[ \frac{4}{6} = \frac{3}{6} + P(B) - \frac{1}{6} \implies \frac{4}{6} = \frac{2}{6} + P(B) \]

### Step 5: Solve for \( P(B) \)

To isolate \( P(B) \):
\[ P(B) = \frac{4}{6} - \frac{2}{6} = \frac{2}{6} = \frac{1}{3} \]

### Step 6: Find \( P(\bar{A} \cap B) \)

We can use the relationship:
\[ P(B) = P(A \cap B) + P(\bar{A} \cap B) \]
Thus,
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]
Substituting the values we found:
\[ P(\bar{A} \cap B) = \frac{1}{3} - \frac{1}{6} \]

### Step 7: Convert to a common denominator and simplify

Convert \( \frac{1}{3} \) to sixths:
\[ \frac{1}{3} = \frac{2}{6} \]
Now, substituting:
\[ P(\bar{A} \cap B) = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \]

### Final Answer

Thus, the probability \( P(\bar{A} \cap B) \) is:
\[ \boxed{\frac{1}{6}} \]