uniform distribution with interval (0,2) and sample 12 
Q): Suppose that you wish to sample $12$ observations randomly from a uniform distribution on the interval $(0,2)$. An approximate value of the probability that the average of your sample will be less than $0.5$.

what i did was i found the $\mathbb E(x)$ to be $1$ and variance to be $1/3$
and then i wasnt sure where to go with this so i decided to do the $z$-formula of 
$(0.5-1)/\sqrt{1/3}/\sqrt{12}$ to get me $z = -3$ then i got $p= 0.001350$ from that , i was just wondering if that was correct.
 A: The expectation of a single sample is $1$, with variance $\frac13$, so the average of twelve samples has an expectation of $1$ and variance of $\frac{1}{36}$, saying (as you found) that for this average $0.5$ would be $3$ standard deviations below the mean.  For a standard normal distribution, you are correct that the probability of values less than $3$ standard deviations below the mean is about $0.00135$ 
The difficult question is whether a normal approximation is a good approximation in this case: in particular normal approximations can sometimes perform relatively poorly in the tails 
Your question actually involves a Bates distribution, but it is easier to handle translating it to the Irwing-Hall distribution: you want the probability that the sum of $12$ uniform i.i.d. random variables on $[0,1]$ is less than $3$.  This is $$\frac{1}{12!}\left({12 \choose 0}(3-0)^{12} - {12 \choose 1}(3-1)^{12} + {12 \choose 2}(3-2)^{12}- {12 \choose 3}(3-3)^{12}\right) \approx 0.001007$$
So the exact answer is rather less than the normal approximation.  Whether the normal approximation is good enough, I leave for you to judge 
