What does this sentence mean translated to probability theory In my exercise sheets I found the sentence:
There will be 12 exercise sheets in this semester. Some errors are found on the sheets by two per ten sheets, on
average.
This is really rather a linguistic than a mathematical problem.
Does it mean that if look at 10 sheets, we will find two sheets that contain an error? I.e. does it mean that $P (\text{error on particular sheet}) = \frac{1}{5}$? 

It means that the expected (mean) number of errors per sheet is $\frac{1}{5}$. That does not mean that the probability of $1$ error, or of at least one error, is equal to $\frac{1}{5}$.
The usual mathematical model is that the number of errors in a randomly chosen sheet has Poisson distribution with parameter $\lambda=\frac{1}{5}$.  It turns out that under this model, the probability of exactly $1$ error in a randomly selected sheet is $\lambda e^{-\lambda}$, and the probability of at least $1$ error is $1-e^{-\lambda}$.
The number of errors in $10$ sheets has Poisson distribution with parameter $2$. The number of errors in $12$ sheets has Poisson distribution with parameter $2\cdot\frac{12}{10}$. 
