Give that `P(A) = 0.5 , P(B) = 0.35 , P(A uu B) = 0.7` find ` P(A nn B)`

Answer

To find \( P(A \cap B) \) (the probability of the intersection of events A and B), we can use the formula for the probability of the union of two events:

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

### Step-by-Step Solution:

1. **Write down the given probabilities:**
- \( P(A) = 0.5 \)
- \( P(B) = 0.35 \)
- \( P(A \cup B) = 0.7 \)

2. **Substitute the known values into the union formula:**
\[ 0.7 = 0.5 + 0.35 - P(A \cap B) \]

3. **Combine the probabilities on the right side:**
\[ 0.7 = 0.85 - P(A \cap B) \]

4. **Rearrange the equation to solve for \( P(A \cap B) \):**
\[ P(A \cap B) = 0.85 - 0.7 \]

5. **Calculate the value:**
\[ P(A \cap B) = 0.15 \]

### Final Answer:
\[ P(A \cap B) = 0.15 \]