Hi @chrsmth . You could think about it this way. Prob(finding at least 1)=Prob(x>0), where x is the number of failures in a sample size of n, = 1 - Prob(x=0)^n.
1 - Prob(x=0)^n = 1-(1-p)^n = 0.95.
This implies (1-p)^n = 0.05.
This implies n = ln(0.05)/ln(1-p) where p = 0.3 or 0.1. Then round n up to the nearest integer.
n is then the sample size such that there is 95% chance of at least one of them is a failure.
The thing is, this is a question of probability, not statistics. Problems of statistics involve testing/estimating parameters based on observed data. In your case, you say you know the parameter p (0.3). So, once p is assumed to be known, the sample size question, as you state it, is all about probability. I.e., you know the distribution is binomial(n, p), where p = 0.3. So, once the distribution is known there are no statistical hypotheses to be tested. It is a matter of probability.
Now…if you want to “prove” (I’m using “prove” very loosely here) that p = 0.3, then that is a problem of statistics.