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

Explanation

To determine whether events E and F are independent, we need to check if the following condition holds:

\[ P(E \cap F) = P(E) \cdot P(F) \]

### Step 1: Identify the given probabilities
We are given:
- \( P(E) = \frac{3}{5} \)
- \( P(F) = \frac{3}{10} \)
- \( P(E \cap F) = \frac{1}{5} \)

### Step 2: Calculate \( P(E) \cdot P(F) \)
Now, we will calculate the product of \( P(E) \) and \( P(F) \):

\[ P(E) \cdot P(F) = \left( \frac{3}{5} \right) \cdot \left( \frac{3}{10} \right) \]

Calculating this gives:

\[ P(E) \cdot P(F) = \frac{3 \cdot 3}{5 \cdot 10} = \frac{9}{50} \]

### Step 3: Compare \( P(E \cap F) \) with \( P(E) \cdot P(F) \)
Now, we compare \( P(E \cap F) \) with \( P(E) \cdot P(F) \):

- We have \( P(E \cap F) = \frac{1}{5} \).
- Converting \( \frac{1}{5} \) to a fraction with a denominator of 50 gives us \( \frac{10}{50} \).

Now we compare:

\[ P(E \cap F) = \frac{10}{50} \quad \text{and} \quad P(E) \cdot P(F) = \frac{9}{50} \]

### Step 4: Conclusion
Since \( P(E \cap F) \neq P(E) \cdot P(F) \), we conclude that:

\[ \frac{10}{50} \neq \frac{9}{50} \]

Therefore, events E and F are **not independent**.

### Summary of Steps:
1. Identify the given probabilities.
2. Calculate the product of \( P(E) \) and \( P(F) \).
3. Compare \( P(E \cap F) \) with \( P(E) \cdot P(F) \).
4. Conclude whether E and F are independent or not.