Let E and F be two events associated with the same random experiment. Then E and F are said to be independent if `P(E nn F)` is equal to :

Options

A. `(P(E))/(P(F))`
B. `P(E)+P(F)`
C. `P(E)-P(F)`
D. `P(E).P(F)`

Answer

To determine the condition for two events \( E \) and \( F \) to be independent, we need to understand the relationship between their probabilities.

### Step-by-Step Solution:

1. **Definition of Independent Events**:
Two events \( E \) and \( F \) are said to be independent if the occurrence of one does not affect the occurrence of the other. Mathematically, this is expressed as:
\[ P(E \cap F) = P(E) \cdot P(F) \]

2. **Understanding the Intersection**:
The intersection \( E \cap F \) represents the event that both \( E \) and \( F \) occur simultaneously. For independent events, the probability of both events occurring together is the product of their individual probabilities.

3. **Using the Definition**:
According to the definition of independence:
\[ P(E \cap F) = P(E) \cdot P(F) \]
This means that if we know the probabilities of \( E \) and \( F \), we can find the probability of both events occurring together by multiplying these probabilities.

4. **Conclusion**:
Therefore, the condition for the independence of events \( E \) and \( F \) can be summarized as:
\[ P(E \cap F) = P(E) \cdot P(F) \]

### Final Answer:
Thus, \( P(E \cap F) \) is equal to \( P(E) \cdot P(F) \) when \( E \) and \( F \) are independent events.

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