This is not a mathematical question. Mathematically speaking, independence is an assumption. That is, we can model the experiment you describe mathematically by several independent events, but nothing mathematically forces us to model the events as independent; the motivation for that comes from extra-mathematical sources, namely a) the fact that this model turns out to be good at predicting instances of this experiment and b) most of us have an overall world view which suggests that events in the physical world occur due to physical causes, and since, as Qiaochu pointed out, no physical mechanism is known or easily imaginable that would make these events depend on each other, the assumption of independence makes physical sense.
None of this, however, proves that the events are actually independent. Whether they are or not is a question for physics, philosophy and religion, not for mathematics. If you believe that there's a karma account and no-one is entitled to more than a fair share of luck, or if you believe that there exists a god who determines whether you win or lose in a dice game, or if you believe that there are morphogenetic fields that cause a tendency for things to happen as they've happened in the past, then all those beliefs imply that the events are in fact not independent, and nothing in mathematics can disprove any of those beliefs.