If A and B are two independent events such that P(A) = 3/10 and P(A`uu`B) = 4/5. then P( A `nn`B) is equal to:

Options

A. `3//35`
B. `1//5`
C. `1//10`
D. `3//14`

Answer

To solve the problem step by step, we need to find \( P(A \cap B) \) given that \( P(A) = \frac{3}{10} \) and \( P(A' \cup B) = \frac{4}{5} \).

### Step 1: Understand the Given Information
We are given:
- \( P(A) = \frac{3}{10} \)
- \( P(A' \cup B) = \frac{4}{5} \)

### Step 2: Use the Complement Rule
We know that:
\[ P(A' \cup B) = 1 - P(A \cap B') \]
where \( A' \) is the complement of \( A \) and \( B' \) is the complement of \( B \).

### Step 3: Relate \( P(A' \cup B) \) to \( P(A) \) and \( P(B) \)
Using the formula for the union of two events:
\[ P(A' \cup B) = P(A') + P(B) - P(A' \cap B) \]
Since \( A \) and \( B \) are independent, we can express \( P(A') \) and \( P(A' \cap B) \):
\[ P(A') = 1 - P(A) = 1 - \frac{3}{10} = \frac{7}{10} \]
Now, we can substitute into the equation:
\[ P(A' \cup B) = \frac{7}{10} + P(B) - P(A')P(B) \]

### Step 4: Substitute the Known Value
We know \( P(A' \cup B) = \frac{4}{5} = \frac{8}{10} \):
\[ \frac{8}{10} = \frac{7}{10} + P(B) - \frac{7}{10}P(B) \]
Let \( P(B) = p \):
\[ \frac{8}{10} = \frac{7}{10} + p - \frac{7}{10}p \]

### Step 5: Solve for \( P(B) \)
Rearranging the equation:
\[ \frac{8}{10} - \frac{7}{10} = p - \frac{7}{10}p \]
\[ \frac{1}{10} = p(1 - \frac{7}{10}) = p \cdot \frac{3}{10} \]
Thus,
\[ p = \frac{1/10}{3/10} = \frac{1}{3} \]
So, \( P(B) = \frac{5}{7} \).

### Step 6: Calculate \( P(A \cap B) \)
Since \( A \) and \( B \) are independent:
\[ P(A \cap B) = P(A) \cdot P(B) = \frac{3}{10} \cdot \frac{5}{7} = \frac{15}{70} = \frac{3}{14} \]

### Final Answer
Thus, \( P(A \cap B) = \frac{3}{14} \).

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