Given `P(A uu B) = 5/6, P(A nn B) = 1/3 and P(bar(B)) = 1/2.` What is `P(bar(A))` ?

Options

A. `1/6`
B. `1/3`
C. `2/3`
D. `1/2`

Explanation

To find \( P(\bar{A}) \), we will use the information provided in the question and apply the relevant probability formulas step by step.

### Step 1: Identify Given Values
We are given:
- \( P(A \cup B) = \frac{5}{6} \)
- \( P(A \cap B) = \frac{1}{3} \)
- \( P(\bar{B}) = \frac{1}{2} \)

### Step 2: Find \( P(B) \)
Using the complement rule:
\[ P(B) = 1 - P(\bar{B}) = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 3: Use the Formula for \( P(A \cup B) \)
We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ \frac{5}{6} = P(A) + \frac{1}{2} - \frac{1}{3} \]

### Step 4: Simplify the Equation
First, convert \( \frac{1}{2} \) and \( \frac{1}{3} \) to a common denominator (which is 6):
\[ \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6} \]
Now substitute these values back into the equation:
\[ \frac{5}{6} = P(A) + \frac{3}{6} - \frac{2}{6} \]
This simplifies to:
\[ \frac{5}{6} = P(A) + \frac{1}{6} \]

### Step 5: Solve for \( P(A) \)
Now, isolate \( P(A) \):
\[ P(A) = \frac{5}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3} \]

### Step 6: Find \( P(\bar{A}) \)
Using the complement rule again:
\[ P(\bar{A}) = 1 - P(A) = 1 - \frac{2}{3} = \frac{1}{3} \]

### Final Answer
Thus, the probability \( P(\bar{A}) \) is:
\[ \boxed{\frac{1}{3}} \]