If `P(overline A cup B) = 5/6, P(A) = 1/2 and P(overlineB) = 2/3," then : " P(Acup B) =`

Options

A. `1/3`
B. `5/6`
C. `2/3`
D. `4/9`

Answer

To solve the problem, we need to find \( P(A \cup B) \) given the following probabilities:

1. \( P(\overline{A} \cup B) = \frac{5}{6} \)
2. \( P(A) = \frac{1}{2} \)
3. \( P(\overline{B}) = \frac{2}{3} \)

### Step-by-step Solution:

**Step 1: Find \( P(B) \)**
We know that \( P(\overline{B}) = \frac{2}{3} \).
Using the complement rule, we can find \( P(B) \):
\[ P(B) = 1 - P(\overline{B}) = 1 - \frac{2}{3} = \frac{1}{3} \]

**Step 2: Use the formula for \( P(A \cup B) \)**
We use the formula for the union of two events:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

**Step 3: Find \( P(A \cap B) \)**
We can express \( P(A \cap B) \) using the information given about \( P(\overline{A} \cup B) \):
\[ P(\overline{A} \cup B) = P(B) + P(\overline{A}) - P(\overline{A} \cap B) \]
Since \( P(\overline{A}) = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} \), we can substitute:
\[ P(\overline{A} \cup B) = P(B) + P(\overline{A}) - P(\overline{A} \cap B) \]
We can also express \( P(\overline{A} \cap B) \) as:
\[ P(\overline{A} \cap B) = P(B) - P(A \cap B) \]
Substituting \( P(B) = \frac{1}{3} \) and \( P(\overline{A} \cup B) = \frac{5}{6} \):
\[ \frac{5}{6} = \frac{1}{3} + \frac{1}{2} - P(A \cap B) \]

**Step 4: Solve for \( P(A \cap B) \)**
Convert \( \frac{1}{3} \) and \( \frac{1}{2} \) to a common denominator (which is 6):
\[ \frac{1}{3} = \frac{2}{6}, \quad \frac{1}{2} = \frac{3}{6} \]
Now substitute these values:
\[ \frac{5}{6} = \frac{2}{6} + \frac{3}{6} - P(A \cap B) \]
This simplifies to:
\[ \frac{5}{6} = \frac{5}{6} - P(A \cap B) \]
Thus, we find:
\[ P(A \cap B) = 0 \]

**Step 5: Calculate \( P(A \cup B) \)**
Now substitute \( P(A) \), \( P(B) \), and \( P(A \cap B) \) into the union formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
\[ P(A \cup B) = \frac{1}{2} + \frac{1}{3} - 0 \]
Convert to a common denominator (which is 6):
\[ P(A \cup B) = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]

### Final Answer:
\[ P(A \cup B) = \frac{5}{6} \]