I've been thinking of something for some time now, and since I am not very proficient in probability theory I thought this could be a good place to ask this question. This is something that came up to me in the long queues of the public transport.
Suppose that you're in a bus station, and you know that a bus (or several buses) will certainly come in the future (during the day,) but you don't know the exact moment. You imagine a probability that the bus will arrive within five minutes. So you wait five minutes. But the bus doesn't arrive. Is now the probability less than or greater than the original one you imagined?
The question is because if you're using the past to predict the future, maybe you won't be very optimistic about the bus arriving. But maybe you could also think that it actually makes the event more likely: since the bus hasn't arrived yet, there are less minutes available in the day and thus the probability is higher.
Think of the last five minutes of the day. You've been there the whole day and no buses have come. So, judging solely from the past, you can't predict that the bus is going to arrive within the next five minutes. But since you're sure that a bus will arrive before the day ends, and there are only five minutes for the day to end, you can be 100% sure that the bus wil arrive within five minutes.
So, the question is, if I'm going to calculate the probability and drop out of the queue, which method should I use? It's because sometimes I quit and suddenly the bus arrives, but sometimes I wait and wait and wait and the bus doesn't come. Or maybe this whole question is nonsense and that is simply terribly random?