If `P(AuuB)=0.65andP(AcapB)=0.15`, find `P(overline(A))+P(overline(B))`.

Answer

To solve the problem, we need to find \( P(\overline{A}) + P(\overline{B}) \) given that \( P(A \cup B) = 0.65 \) and \( P(A \cap B) = 0.15 \).

### Step-by-Step Solution:

1. **Understand the Complement Rule**:
The probability of the complement of an event \( A \) is given by:
\[ P(\overline{A}) = 1 - P(A) \]
Similarly,
\[ P(\overline{B}) = 1 - P(B) \]

2. **Use the Formula for Union**:
We know from probability theory that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We can rearrange this to find \( P(A) + P(B) \):
\[ P(A) + P(B) = P(A \cup B) + P(A \cap B) \]

3. **Substitute the Given Values**:
Substitute the known values into the equation:
\[ P(A) + P(B) = 0.65 + 0.15 = 0.80 \]

4. **Calculate \( P(\overline{A}) + P(\overline{B}) \)**:
Now, we can find \( P(\overline{A}) + P(\overline{B}) \):
\[ P(\overline{A}) + P(\overline{B}) = (1 - P(A)) + (1 - P(B)) \]
This simplifies to:
\[ P(\overline{A}) + P(\overline{B}) = 2 - (P(A) + P(B)) \]
Substitute \( P(A) + P(B) = 0.80 \):
\[ P(\overline{A}) + P(\overline{B}) = 2 - 0.80 = 1.20 \]

### Final Answer:
Thus, the value of \( P(\overline{A}) + P(\overline{B}) \) is \( 1.20 \).

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