Let E and F be events with :
`P(E)=(4)/(5),P(F)=(3)/(10)` and `P(E nn F)=(1)/(5)`. Are E and F independent ?

Answer

To determine whether the events E and F are independent, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- \( P(E) = \frac{4}{5} \)
- \( P(F) = \frac{3}{10} \)
- \( P(E \cap F) = \frac{1}{5} \)

2. **Recall the Definition of Independence:**
- Two events E and F are independent if:
\[ P(E \cap F) = P(E) \times P(F) \]

3. **Calculate \( P(E) \times P(F) \):**
- Calculate the product of the probabilities:
\[ P(E) \times P(F) = \left(\frac{4}{5}\right) \times \left(\frac{3}{10}\right) \]
- Multiply the fractions:
\[ = \frac{4 \times 3}{5 \times 10} = \frac{12}{50} \]
- Simplify \( \frac{12}{50} \):
\[ = \frac{6}{25} \]

4. **Compare \( P(E \cap F) \) with \( P(E) \times P(F) \):**
- We have:
- \( P(E \cap F) = \frac{1}{5} \)
- \( P(E) \times P(F) = \frac{6}{25} \)

5. **Convert \( P(E \cap F) \) to a common denominator:**
- Convert \( \frac{1}{5} \) to have a denominator of 25:
\[ \frac{1}{5} = \frac{5}{25} \]

6. **Compare the two values:**
- Now we compare:
\[ P(E \cap F) = \frac{5}{25} \quad \text{and} \quad P(E) \times P(F) = \frac{6}{25} \]
- Since \( \frac{5}{25} \neq \frac{6}{25} \), we conclude that:
\[ P(E \cap F) \neq P(E) \times P(F) \]

7. **Conclusion:**
- Since the condition for independence is not satisfied, we conclude that events E and F are **not independent**.