For two events A and B. `P(A)= (1)/(2)` let `P(A uu B)= (2)/(3)` and `P(Ann B)= (1)/(6)`. What is P `(barA nnB)` equal to ?

Options

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

Answer

To solve for \( P(\bar{A} \cap B) \), we can follow these steps:

### Step 1: Understand the given probabilities
We are given:
- \( P(A) = \frac{1}{2} \)
- \( P(A \cup B) = \frac{2}{3} \)
- \( P(A \cap B) = \frac{1}{6} \)

### Step 2: Use the formula for the union of two events
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values into this formula:
\[ \frac{2}{3} = \frac{1}{2} + P(B) - \frac{1}{6} \]

### Step 3: Solve for \( P(B) \)
Rearranging the equation to isolate \( P(B) \):
\[ P(B) = \frac{2}{3} + \frac{1}{6} - \frac{1}{2} \]

To perform the addition and subtraction, we need a common denominator. The least common multiple of 3, 6, and 2 is 6. Thus, we convert each term:
- \( \frac{2}{3} = \frac{4}{6} \)
- \( \frac{1}{2} = \frac{3}{6} \)

Now substituting these values back:
\[ P(B) = \frac{4}{6} + \frac{1}{6} - \frac{3}{6} = \frac{4 + 1 - 3}{6} = \frac{2}{6} = \frac{1}{3} \]

### Step 4: Calculate \( P(\bar{A} \cap B) \)
We know that:
\[ P(\bar{A} \cap B) = P(B) - P(A \cap B) \]
Substituting the values we found:
\[ P(\bar{A} \cap B) = \frac{1}{3} - \frac{1}{6} \]

### Step 5: Solve for \( P(\bar{A} \cap B) \)
To subtract these fractions, we convert \( \frac{1}{3} \) to have a denominator of 6:
\[ \frac{1}{3} = \frac{2}{6} \]
Now substituting:
\[ P(\bar{A} \cap B) = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \]

### Final Answer
Thus, \( P(\bar{A} \cap B) = \frac{1}{6} \).

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