How to test component failure in a general way? Let's say I have 100 parts. I sample one part and find it defective. I then sample another five parts and do not find them defective. What statistical procedure would I use that combines this information to provide a confidence interval or failure rate?
 A: Let's assume you choose the parts at random, and that you are sampling without replacement so all 6 parts are different.  
So you now know that 1 out of 6 parts sampled were defective.  You can be sure that between 1 and 95 of your 100 parts are defective;  you probably suspect that the actual number is closer to the bottom end.
To get a better estimate you can use Bayesian methods, with a prior belief about the number of defective parts.  Perhaps you might have started by thinking that any number from 0 through to 100 are defective and each number was equally likely.  If the number actually defective is $N$ out of 100, then the likelihood of 1 defective in 6 tested is proportional to ${N \choose 1}{100-N \choose 5}$ and so the posterior probability is this divided by its sum over $N$, which is ${101 \choose 7}$.  This is a distribution for $N$, with a mode at $N=17$, a median at $N=22$, a mean of $24.5$, a variance of $199.75$, a standard deviation about $14.13$, and a Bayesian 95% credible interval of something like $[2,52]$.   
To tighten this range you need to test more, or to have started with a tighter prior distribution for your beliefs about the number of defectives.
