Ten tasters and a 10 percent chance at the poison

Exactly one of $n$ wine barrels is poisoned. A king has $10$ tasters, and anyone who sips from the poisoned barrel dies exactly one month later; sipping from safe barrels is harmless. The tasters will only take part for a single month, so there is just one round of tasting. What is the largest $n$ for which the king can guarantee at least a $10\%$ probability of naming the poisoned barrel?

10240. The set of tasters who die can be any of $2^{10}=1024$ subsets, so give each subset its own group of $10$ barrels (each barrel tasted by exactly the tasters in its subset). The death pattern then pins the poison to one group of $10$, and a random guess within it succeeds with probability $\tfrac{1}{10}$; larger $n$ forces some pattern to cover more than $10$ barrels. Hence $n=10\cdot 2^{10}$.