IF A and B are two indepenent events such that `P( barA ) = 0.65 , P (A uu B ) =0.65` and ` P(B ) =p,` find the value of p.

Options

A. `7/13`
B. `6/13`
C. `37/65`
D. none of these

Answer

To solve the problem step by step, we need to find the value of \( p \) given the probabilities of independent events \( A \) and \( B \).

### Step 1: Understand the given probabilities
We have:
- \( P(\bar{A}) = 0.65 \) (the probability of the complement of event \( A \))
- \( P(A \cup B) = 0.65 \) (the probability of the union of events \( A \) and \( B \))
- \( P(B) = p \) (the probability of event \( B \))

### Step 2: Calculate \( P(A) \)
Since \( P(\bar{A}) + P(A) = 1 \), we can find \( P(A) \):
\[ P(A) = 1 - P(\bar{A}) = 1 - 0.65 = 0.35 \]

### Step 3: Use the formula for the probability of the union of two independent events
The formula for the probability of the union of two independent events \( A \) and \( B \) is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Since \( A \) and \( B \) are independent, we have:
\[ P(A \cap B) = P(A) \cdot P(B) = P(A) \cdot p \]
Substituting this into the union formula gives:
\[ P(A \cup B) = P(A) + P(B) - P(A) \cdot P(B) \]

### Step 4: Substitute known values into the equation
Now we can substitute the known values into the equation:
\[ 0.65 = 0.35 + p - (0.35 \cdot p) \]

### Step 5: Simplify the equation
Rearranging the equation gives:
\[ 0.65 = 0.35 + p - 0.35p \]
Combining like terms results in:
\[ 0.65 = 0.35 + p(1 - 0.35) \]
This simplifies to:
\[ 0.65 = 0.35 + 0.65p \]

### Step 6: Isolate \( p \)
To isolate \( p \), we subtract \( 0.35 \) from both sides:
\[ 0.65 - 0.35 = 0.65p \]
This simplifies to:
\[ 0.30 = 0.65p \]

### Step 7: Solve for \( p \)
Now, we can solve for \( p \):
\[ p = \frac{0.30}{0.65} \]
To simplify this fraction:
\[ p = \frac{30}{65} = \frac{6}{13} \]

### Final Answer
Thus, the value of \( p \) is:
\[ \boxed{\frac{6}{13}} \]