If P(B)`=3/4,P(A cap B cap overset -c)1/3`and `P(overset -A cap B cap overset -C)1/3`,then `P(B cap C)` is

Options

A. `1//12`
B. `1//6`
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 it down step by step.

### Step 1: Write down the given probabilities
We have:
- \( P(B) = \frac{3}{4} \)
- \( P(A \cap B \cap C') = \frac{1}{3} \)
- \( P(A' \cap B \cap C') = \frac{1}{3} \)

### Step 2: Use the formula for \( P(B \cap C) \)
We can use the following formula to find \( P(B \cap C) \):

\[ P(B \cap C) = P(B) - P(A \cap B \cap C') + P(A' \cap B \cap C') \]

### Step 3: Substitute the values into the formula
Substituting the values we have:

\[ P(B \cap C) = \frac{3}{4} - \frac{1}{3} + \frac{1}{3} \]

### Step 4: Simplify the expression
Notice that \( -\frac{1}{3} + \frac{1}{3} = 0 \), so we have:

\[ P(B \cap C) = \frac{3}{4} - 0 = \frac{3}{4} \]

### Step 5: Calculate the final value
Now we need to ensure we have the correct value for \( P(B \cap C) \):
- Convert \( \frac{3}{4} \) to a fraction with a common denominator to combine with \( -\frac{1}{3} \).

Finding the least common multiple (LCM) of 4 and 3, which is 12:

\[ P(B) = \frac{3}{4} = \frac{9}{12} \]
\[ P(A \cap B \cap C') = \frac{1}{3} = \frac{4}{12} \]
\[ P(A' \cap B \cap C') = \frac{1}{3} = \frac{4}{12} \]

Now substituting these values back into the formula:

\[ P(B \cap C) = \frac{9}{12} - \frac{4}{12} + \frac{4}{12} \]

This simplifies to:

\[ P(B \cap C) = \frac{9}{12} - \frac{4}{12} + \frac{4}{12} = \frac{9}{12} \]

### Step 6: Final simplification
Thus, we can simplify \( \frac{9}{12} \) to:

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

### Final Answer
Therefore, the value of \( P(B \cap C) \) is \( \frac{3}{4} \).

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