The question in the image asks about independent events and their properties. Here's a breakdown of the answer choices:

1. a) "cannot happen at the same time" - This describes mutually exclusive events, not independent events. Mutually exclusive events cannot occur together, while independent events can.

2. b) "mean that P(A|B) = P(A)" - This is the correct definition of independent events. Independence implies that the occurrence of event B does not affect the probability of event A, which is mathematically expressed as $P(A|B) = P(A)$.

3. c) "mean that P(A and B) = 0" - This is also describing mutually exclusive events. If two events are mutually exclusive, they cannot happen together, so $P(A \cap B) = 0$. However, independent events can happen simultaneously, so this is incorrect.

4. d) "require that one event occur before the other can happen" - This suggests a temporal or conditional relationship, which does not apply to independent events. Independent events can occur in any order or at the same time.

Correct Answer: b) "mean that P(A|B) = P(A)"

Would you like further details or have any questions? Here are five related questions to expand your understanding:

1. What is the difference between independent events and mutually exclusive events?
2. Can two independent events also be mutually exclusive?
3. How would you calculate the probability of two independent events both occurring?
4. What is the significance of conditional probability in relation to independent events?
5. How can you determine if two events are independent from a dataset?

Tip: For two independent events $A$ and $B$, the probability of both events happening is given by $P(A \cap B) = P(A) \times P(B)$.