Let A and B be independent events. Denote P(A) as the probability that Event A will occur, and denote `P(A nn B)` as the probability that Events A and B will both occur. Which of the following equations must be true?

Options

A. `P(A) = P(B)`
B. `P(A) = 1 - P(B)`
C. `P(A nn B) = P(A) + P(B)`
D. `P(A nn B) = P(A) cdot P(B)`

Answer

To solve the problem regarding independent events A and B, we need to understand the relationships between their probabilities.

### Step-by-Step Solution:

1. **Understanding Independent Events**:
- Two events A and B are said to be independent if the occurrence of one does not affect the occurrence of the other.

2. **Probability of Intersection**:
- For independent events, the probability that both events A and B occur (denoted as P(A ∩ B)) is given by the formula:
\[ P(A \cap B) = P(A) \times P(B) \]
- This means that the probability of both events happening together is the product of their individual probabilities.

3. **Reviewing Other Probability Formulas**:
- The probability of either event A or event B occurring (denoted as P(A ∪ B)) is given by:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]
- However, for independent events, this does not simplify to a direct equation between P(A) and P(B).

4. **Analyzing the Given Options**:
- **Option A**: P(A) = P(B)
- This is not necessarily true; independence does not imply equality.
- **Option B**: P(A) = 1 - P(B)
- This is true only for mutually exclusive events, which is not the case here.
- **Option C**: P(A ∩ B) = P(A) + P(B)
- This is incorrect as it does not apply to independent events.
- **Option D**: P(A ∩ B) = P(A) × P(B)
- This is the correct equation for independent events.

5. **Conclusion**:
- The equation that must be true for independent events A and B is:
\[ P(A \cap B) = P(A) \times P(B) \]
- Therefore, the correct answer is Option D.