A and B are two mutually exclusive events of an experiment: If P(not A) = 0.65, `P(A cup B) = 0.65` and P(B) = p, then the value of p is

Options

A. `0.35`
B. `0.25`
C. `0.30`
D. `0.40`

Answer

To solve the problem step by step, we will use the information given about the events A and B.

### Step 1: Understand the given information
We know:
- \( P(\text{not } A) = 0.65 \)
- \( P(A \cup B) = 0.65 \)
- \( P(B) = p \)
- A and B are mutually exclusive events.

### Step 2: Calculate \( P(A) \)
Since \( P(\text{not } A) = 0.65 \), we can find \( P(A) \) using the formula:
\[ P(A) = 1 - P(\text{not } A) \]
Substituting the value:
\[ P(A) = 1 - 0.65 = 0.35 \]

### Step 3: Use the formula for the union of two events
For mutually exclusive events, the probability of their union is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since A and B are mutually exclusive, \( P(A \cap B) = 0 \). Therefore, the formula simplifies to:
\[ P(A \cup B) = P(A) + P(B) \]

### Step 4: Substitute the known values into the formula
We know \( P(A \cup B) = 0.65 \) and \( P(A) = 0.35 \). Substituting these values into the equation:
\[ 0.65 = 0.35 + P(B) \]

### Step 5: Solve for \( P(B) \)
Rearranging the equation to find \( P(B) \):
\[ P(B) = 0.65 - 0.35 \]
\[ P(B) = 0.30 \]

### Conclusion
Thus, the value of \( p \) is:
\[ \boxed{0.30} \]