If `P(B)= (3)/(4), P(A nn B nn barC)= (1)/(3)` and `(barA nnB nn bar C) =(1)/(3)` then what is `P (B nnC)` equals to ?

Options

A. `1/(12)`
B. `3/4`
C. `1/(15)`
D. `1/9`

Answer

To solve the problem, we need to find \( P(B \cap C) \) using the given probabilities. Let's break down the solution step by step.

### Step 1: Understand the Given Probabilities
We have the following probabilities:
- \( P(B) = \frac{3}{4} \)
- \( P(A \cap B \cap \bar{C}) = \frac{1}{3} \)
- \( P(\bar{A} \cap B \cap \bar{C}) = \frac{1}{3} \)

### Step 2: Calculate \( P(B \cap \bar{C}) \)
To find \( P(B \cap C) \), we first need to find \( P(B \cap \bar{C}) \). This can be calculated by adding the probabilities of the two intersections involving \( B \) and \( \bar{C} \):

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

Substituting the values we have:

\[ P(B \cap \bar{C}) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \]

### Step 3: Use the Total Probability of B
Now, we can use the total probability of \( B \) to find \( P(B \cap C) \):

\[ P(B) = P(B \cap C) + P(B \cap \bar{C}) \]

Rearranging this gives us:

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

### Step 4: Substitute the Values
Now we substitute the known values into the equation:

\[ P(B \cap C) = \frac{3}{4} - \frac{2}{3} \]

### Step 5: Find a Common Denominator
To perform the subtraction, we need a common denominator. The least common multiple of 4 and 3 is 12.

Converting the fractions:

\[ \frac{3}{4} = \frac{9}{12} \]
\[ \frac{2}{3} = \frac{8}{12} \]

### Step 6: Perform the Subtraction
Now we can subtract:

\[ P(B \cap C) = \frac{9}{12} - \frac{8}{12} = \frac{1}{12} \]

### Final Answer
Thus, the probability \( P(B \cap C) \) is:

\[ \boxed{\frac{1}{12}} \]