If `P(A nnB)=1/2` and `P(A' nn B')=1/3, P(A)=2p` and P(B)=p , then the value of 9p is ____

Answer

To solve the problem step by step, we will use the given probabilities and relationships between events A and B.

### Step 1: Write down the given information
We know:
- \( P(A \cap B) = \frac{1}{2} \)
- \( P(A' \cap B') = \frac{1}{3} \)
- \( P(A) = 2p \)
- \( P(B) = p \)

### Step 2: Use De Morgan's Law
According to De Morgan's Law:
\[ P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B) \]
Thus, we can write:
\[ P(A' \cap B') = 1 - P(A \cup B) \]

### Step 3: Substitute the known value
From the second piece of information, we have:
\[ \frac{1}{3} = 1 - P(A \cup B) \]
This implies:
\[ P(A \cup B) = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Use the formula for the probability of the union of two events
The formula for the probability of the union of two events is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ P(A \cup B) = 2p + p - \frac{1}{2} \]
This simplifies to:
\[ P(A \cup B) = 3p - \frac{1}{2} \]

### Step 5: Set the two expressions for \( P(A \cup B) \) equal to each other
Now we have two expressions for \( P(A \cup B) \):
\[ \frac{2}{3} = 3p - \frac{1}{2} \]

### Step 6: Solve for \( p \)
To solve for \( p \), first add \( \frac{1}{2} \) to both sides:
\[ \frac{2}{3} + \frac{1}{2} = 3p \]
To add these fractions, find a common denominator (which is 6):
\[ \frac{4}{6} + \frac{3}{6} = 3p \]
This gives:
\[ \frac{7}{6} = 3p \]

### Step 7: Isolate \( p \)
Now, divide both sides by 3:
\[ p = \frac{7}{18} \]

### Step 8: Calculate \( 9p \)
Now, we need to find \( 9p \):
\[ 9p = 9 \times \frac{7}{18} = \frac{63}{18} = \frac{7}{2} = 3.5 \]

### Final Answer
Thus, the value of \( 9p \) is \( \frac{7}{2} \) or \( 3.5 \).

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