IIRC: For repeated trials, the chance of getting exactly one positive result (to account for the cooldown, in this case) would be Pn = (Pproc)*n - (Pproc)^n. From there, you set the desired value for Pn and solve for n. Of course, the algebra for this particular formula is not fun. So, if you were doing it by hand, you'd just plug in successive values of n, starting with a rough guess.

Or, if you're lazy and on the internet, you just ask Wolfram-Alpha to solve the above for a Pn of 0.95 and get a result of 6.33337. Round up to 7, of course, because we're using discrete trials.

That's the sort of hacky I'm-not-a-real-mathematician way. For the real method, see

here. Short version is that running successive, independent trials until a desired result is achieved is a geometric random variable, and the number of trials needed to achieve x "successes" given a probability of success p is simply x/p. In this case, 1/.15 or 6.66... Practically speaking, again because we're talking about discrete trials, treat this as 7.

(For the curious: The two answers are different because one used what's effectively a 95% confidence level, while the other used a discrete calculation. Ultimately, 0.95/.15 is 6.33.... Don't ask me why the Wolfram-Alpha equation solver spat out something non-repeating, though.)