What does it mean for two events to be independent?

"We’ve seen that a particular problem can involve multiple different events. In the All-time Idol example, we considered the probability of a female winner, a country singer winner, and an underage winner, among other things. Now one question that often arises concerns the independence of events. Two events A and B are called independent if the prior probability is the same as the conditional probability; that is, if Pr( A | B ) = Pr( A ). If you reflect on what this means, you’ll see that with independent events, knowing that one of them occurred tells you nothing (either for or against) about whether the other one also occurred. For example, let S be the event that Strike For Gold wins the Kentucky Derby next May. Let R be the event that it rains that day. If I say that S and R are independent, I’m claiming that rain (or the absence thereof) would have no impact either way on the horse’s chances. If you were able to see the future, and reveal to me the weather on Derby Day, that’s fine but it wouldn’t help me in my betting. Knowing Pr( R ) wouldn’t give me any helpful information, because Pr( S | R ) is the same as just plain old Pr( S ) anyway. That’s a conceptual explanation. In the end, it boils down to numbers."