I thought about it for 5 minutes and couldn't find an answer (I found several, but they were all half-answers). I looked at the answer below, and it turned out that you'd missed an important part: The prisoners don't die immediately after drinking from a poisoned barrel... They die after 30 days (in other words, they can drink from several barrels, and if any of those barrels were poisoned, they'd die).

But thanks anyway.