A and B are two events such that `P(A)=3/5, P(B)=3/10` and `P(AuuB)=1/2`
Find whether A and B are independent or not.

Explanation

To determine whether events A and B are independent, we can follow these steps:

### Step 1: Identify the given probabilities
We have:
- \( P(A) = \frac{3}{5} \)
- \( P(B) = \frac{3}{10} \)
- \( P(A \cup B) = \frac{1}{2} \)

### Step 2: Use the formula for independent events
For events A and B to be independent, the following condition must hold:
\[ P(A \cap B) = P(A) \cdot P(B) \]
Calculating \( P(A) \cdot P(B) \):
\[ P(A) \cdot P(B) = \frac{3}{5} \cdot \frac{3}{10} = \frac{9}{50} \]

### Step 3: Use the formula for the union of two events
We know that:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
Substituting the known values:
\[ \frac{1}{2} = \frac{3}{5} + \frac{3}{10} - P(A \cap B) \]

### Step 4: Find a common denominator and simplify
The common denominator for \( \frac{3}{5} \) and \( \frac{3}{10} \) is 10. We convert \( \frac{3}{5} \) to tenths:
\[ \frac{3}{5} = \frac{6}{10} \]
Now substituting back:
\[ \frac{1}{2} = \frac{6}{10} + \frac{3}{10} - P(A \cap B) \]
This simplifies to:
\[ \frac{1}{2} = \frac{9}{10} - P(A \cap B) \]

### Step 5: Solve for \( P(A \cap B) \)
Rearranging gives:
\[ P(A \cap B) = \frac{9}{10} - \frac{1}{2} \]
Finding a common denominator for \( \frac{9}{10} \) and \( \frac{1}{2} \) (which is 10):
\[ \frac{1}{2} = \frac{5}{10} \]
Thus:
\[ P(A \cap B) = \frac{9}{10} - \frac{5}{10} = \frac{4}{10} = \frac{2}{5} \]

### Step 6: Compare \( P(A \cap B) \) with \( P(A) \cdot P(B) \)
Now we compare:
- \( P(A \cap B) = \frac{2}{5} \)
- \( P(A) \cdot P(B) = \frac{9}{50} \)

To compare these, we can convert \( \frac{2}{5} \) to a fraction with a denominator of 50:
\[ \frac{2}{5} = \frac{20}{50} \]

### Step 7: Conclusion
Since \( P(A \cap B) = \frac{20}{50} \) is not equal to \( P(A) \cdot P(B) = \frac{9}{50} \), we conclude that events A and B are **not independent**.