That's dependent on how many transacitons, how many nodes, and the amount 1BTC is worth.
If the fee is uber low, there are many transactions, not as many nodes, and 1BTC is with 1USD, then the likelyhood of a timely transaction is low.
The less demand (transactions) and more supply (nodes), the more likely it will be processed. However due to the low fee (priority) it will be processed at a later time depending on the other two factors as well
If it does work that way...
This matches my understanding.
I think once there are enough nodes/transactions, priority should be affected by the time passed from the transaction as well. You could have a transaction with 0BTC fee occur yesterday, but it would have a higher priority than a transaction with a 1BTC fee made 5 minutes ago.
The problem is that "priority" is determined by the incentives of the miners. A miner has no reason to include a 0BTC fee transaction, even if it's from yesterday, since he gains nothing from it (and presumably including extra transactions reduces his hashing rate, I'm a bit sketchy on that part). There will be calculators that tell you what fee you need to give to gain a certain expected processing time, and the sender should just give that instead of leeching. I suppose it's possible in theory to introduce a "bargaining" feature where the sender allows the offered transaction fee to increase automatically from its initial setting if a lot of time passes with no confirmation.