If a random sample all came out positive, what can be inferred about the population? Specifically, let's say I take a random sample of 20 products from a manufacturing batch of 1000 and they all tested good, what assumptions and conclusions can I make about the whole batch? Is it possible to say "There is an x% chance that the entire batch is good"? How could I calculate x in this case? Note that this is assuming I have no prior knowledge of what the defect rate should be.
 A: 20 of 1000 are good. For rest 980, number of good ones can be from 0 to 980.
Calculate the probabilities that a random sample of 20 products from a manufacturing batch of 1000 and they all tested good, when # of good ones are 20, 21, ..., 1000 among 1000. (totally 981 probabilities). Add them together as denominator, and last one (1000 goods among 1000) as numerator. This ratio is your x.     
Let $Y$ be the number of good ones among 1000 products. Because no prior information, we assume $\Pr(Y=k) = 1/1001$ because $Y$ can be 0, 1, ..., 1000. It is uniform distribution and is used as no informative prior very often.   
Let $B$ be the event that all of 20 products are good in the random sample of 20 from 1000 products. 
So the asked question is
$\Pr(Y=1000|B)$ 
So $\Pr(Y=1000|B) = \frac{Pr(B|Y=1000)\Pr(Y=1000)}{\sum_{y=0}^{1000}\Pr(B|Y=y)\Pr(Y=y)}$ 
$=\frac{1/1001}{1/1001}\frac{Pr(B|Y=1000)}{\sum_{y=0}^{1000}\Pr(B|Y=y)}$
$=\frac{Pr(B|Y=1000)}{\sum_{y=0}^{1000}\Pr(B|Y=y)}$
$=1/47.666667 = 0.02097902$  appr. 2.1%
Under the assumption of simple random sample, the # of good ones among 20 sampled units follows hyper-geometric distribution. So
$\Pr(B|Y=y)={\frac {{\binom {y}{20}}{\binom {980}{0}}}{\binom {1000}{20}}} = \frac {{\binom {y}{20}}}{\binom {1000}{20}}$. Need to know that $\binom {y}{20}= 0 $ for $y<20$
In fact, calculation can be performed by any software with probability density function of hyper-geometric distribution. 
A: You can construct a confidence interval for the proportion $p$ of not defectives.  The interval will be one-sided $[p_c, 1]$ say 95% this gives you an idea based on the width of the interval $([1-p_c,1])$ how confident you are that the proportion is close to 100%. This would be based on the hypergeometric distribution.
Based on Bill Huber's comment about the Bayesian approach.  This shows that a lot can be said without prior knowledge. This computation is based strictly on frequentist methods.
