Probability/ Counting question about Error Rates This is probably a simple question to most of you but I wasn't seeing a clear solution.  I was programming the other day and in one part there is a batch insert into a database of about 30 items at a time.  If one member of the batch has an error then the whole batch will fail.  I found that about 42% of the batches were indeed failing and wondered how I could use this to detect the actual error rate in the dataset.
I fixed the technical aspects of the problem I was facing but I was wondering how to solve problems like this in general.  Abstracted a little it is: if you have sets of x elements (taken at random from the whole dataset) and you know the percent of these sets that have at least one element with a certain property, how can you find roughly the percent of elements in the whole dataset that have that property?
 A: Under certain assumptions, which may or may not be realistic, we can solve the problem.  Assume independence, that is, that success/failure of elements of the database are independent events, like the results of tossing a fair die.  Let $p$ be the (unknown) probability that one individual item fails.
Suppose that the probability that a batch of $k$ items fails is $f$.  We express $f$ in terms of $p$, and then solve for $p$.
The probability $f$ that the batch fails is $1$ minus the probability that all the items in the batch are OK. So
$$f=1-(1-p)^k.$$
We solve for $p$. Manipulation gives 
$$1-f=(1-p)^k.$$
Take the logarithms of both sides, to any base you like.  We will use base $10$.
We obtain
$$\log(1-f)=k\log(1-p),$$
and therefore
$$\log(1-p)=\frac{\log(1-f)}{k}.$$
Raise $10$ to the power of each side. Then
$$10^{\log(1-p)}=1-p=10^{\frac{\log(1-f)}{k}}.$$
It follows that
$$p=1-10^{\frac{\log(1-f)}{k}}.$$
Equivalently, we could use the natural logarithm $\ln$. If we do that, we get
$$p=1-e^{\frac{\ln(1-f)}{k}}.$$
With $f=0.42$, and $k=30$, I get that $p$ is approximately $0.018$.  
