Let A and B be two events with `P(A) = (1)/(3), P(B)= (1)/(6)` and `P(A nnB)= (1)/(12)` What is `P(B|barA)` equal to?

Options

A. `(1)/(5)`
B. `(1)/(7)`
C. `(1)/(8)`
D. `(1)/(10)`

Answer

To find \( P(B | \bar{A}) \), we can use the formula for conditional probability:

\[ P(B | \bar{A}) = \frac{P(B \cap \bar{A})}{P(\bar{A})} \]

### Step 1: Calculate \( P(\bar{A}) \)

Given \( P(A) = \frac{1}{3} \), we can find \( P(\bar{A}) \) as follows:

\[ P(\bar{A}) = 1 - P(A) = 1 - \frac{1}{3} = \frac{2}{3} \]

**Hint**: Remember that the probability of the complement of an event is equal to 1 minus the probability of the event itself.

### Step 2: Calculate \( P(B \cap \bar{A}) \)

We can find \( P(B \cap \bar{A}) \) using the formula:

\[ P(B \cap \bar{A}) = P(B) - P(A \cap B) \]

Given \( P(B) = \frac{1}{6} \) and \( P(A \cap B) = \frac{1}{12} \), we can substitute these values:

\[ P(B \cap \bar{A}) = P(B) - P(A \cap B) = \frac{1}{6} - \frac{1}{12} \]

To perform this subtraction, we need a common denominator (which is 12):

\[ \frac{1}{6} = \frac{2}{12} \]

Now, substituting:

\[ P(B \cap \bar{A}) = \frac{2}{12} - \frac{1}{12} = \frac{1}{12} \]

**Hint**: When subtracting fractions, make sure to convert them to a common denominator.

### Step 3: Substitute into the conditional probability formula

Now we can substitute \( P(B \cap \bar{A}) \) and \( P(\bar{A}) \) into the conditional probability formula:

\[ P(B | \bar{A}) = \frac{P(B \cap \bar{A})}{P(\bar{A})} = \frac{\frac{1}{12}}{\frac{2}{3}} \]

### Step 4: Simplify the fraction

To simplify \( \frac{\frac{1}{12}}{\frac{2}{3}} \), we multiply by the reciprocal of the denominator:

\[ P(B | \bar{A}) = \frac{1}{12} \times \frac{3}{2} = \frac{3}{24} = \frac{1}{8} \]

### Final Answer

Thus, the value of \( P(B | \bar{A}) \) is:

\[ \boxed{\frac{1}{8}} \]