The probability that atleast one of the two events A and B occurs is 0.6. If A and B occur simultaneously with probability 0.3, then evaluate `P (bar A) + P(bar B)` .

Answer

To solve the problem, we need to find \( P(\bar{A}) + P(\bar{B}) \) given that \( P(A \cup B) = 0.6 \) and \( P(A \cap B) = 0.3 \).

### Step 1: Use the formula for the union of two events.
The probability of the union of two events can be expressed as:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
We know:
- \( P(A \cup B) = 0.6 \)
- \( P(A \cap B) = 0.3 \)

### Step 2: Substitute the known values into the formula.
Substituting the known values into the union formula, we get:
\[ 0.6 = P(A) + P(B) - 0.3 \]

### Step 3: Rearrange the equation to find \( P(A) + P(B) \).
Adding \( 0.3 \) to both sides gives:
\[ P(A) + P(B) = 0.6 + 0.3 = 0.9 \]

### Step 4: Find \( P(\bar{A}) + P(\bar{B}) \).
Using the complement rule, we know:
\[ P(\bar{A}) = 1 - P(A) \quad \text{and} \quad P(\bar{B}) = 1 - P(B) \]
Thus,
\[ P(\bar{A}) + P(\bar{B}) = (1 - P(A)) + (1 - P(B)) = 2 - (P(A) + P(B)) \]

### Step 5: Substitute \( P(A) + P(B) \) into the equation.
Now substituting \( P(A) + P(B) = 0.9 \) into the equation:
\[ P(\bar{A}) + P(\bar{B}) = 2 - 0.9 = 1.1 \]

### Final Answer:
Thus, the value of \( P(\bar{A}) + P(\bar{B}) \) is:
\[ \boxed{1.1} \]