The correct option is the first one. Independent events are exactly what you're asking for: the information you have on one event doesn't affect in any way the probability of the other.
A classic example is the roll of two dice: assume that you roll the first one and get a 4. Does this imply anything on the outcome of the second die? No, all events 1,2,3,4,5,6 are still equally distributed, no matter the result of the first die.
