There is a vault containing (N) 10,000 gold bars.
A number of different bars (I) can be chosen at random and inspected for purity. Assume all inspections come back indicating the bar inspected is pure.
How many inspections are needed to be able to say there is a 90% chance that 99.9% of the gold bars are pure?
Let $M$ represent the number of impure golden bars in the vault. The relevant probability statement is $Pr[No\ impure\ bars\ after\ I\ inspections|M\leq10]\geq0.9$. Is there some way to simplify the left side of the equation in terms of the given variables? Then you should be able to solve for $I$.