Probability is about knowledge. It is a measure of how certain you can be that a proposition is true, from your point of view, based on all of your knowledge. We assign a value of 0 to mean "I am certain this is untrue" and 1 to "I am certain this is true", and other values in between.
Note that this means there is no such thing as the probability of an event--only the probability of an event from some observer's point of view. You can see this trivially by noting that if you fairly shuffle a deck, but before dealing out a hand of poker you give the deck to someone to look through before you deal. Now, as you deal the hand, the probabilities of events for the players will play out normally. You should still count your pot odds for that draw the same way. But there's clearly no bet you can offer the man who looked at the cards--from his point of view, all events are either 0 or 1. He has complete knowledge.
Let's say I fairly shuffle a deck¸ and then say "For a $1 bet, I'll give you $50 if the top card of the deck is the Ace of Spades". You should decline that bet, because the probability is 1/52. Now I deal off the top 10 cards into a separate pile, and offer the same bet. Now there are only 42 cards in the deck, but the probability from your point of view is still 1/52. If we repeated this scenario a million times, and you made the second bet each time, you'd win about 1,000,000/52 times, despite the fact you were betting on a 42-card stack. There are still 52 cards unknown to you--it makes no difference if they are in the main pack, across the table, or out the window.
Now let's say I take that pile of 10 cards I dealt off and turn them over, and none of them is the Ace of Spades. Now you should take the bet, because you have new information. The position of the Ace in the pack didn't change, but you have received new information about where it isn't, and that affects your probability.