sum of rounding errors My tax return involves 32 different numbers, each rounded to the nearest dollar and then added together. Assuming that the errors by rounding are uniformly distributed on the interval (-1/2,1/2), estimate the probability that the sum of the rounded amounts differs from the true sum by less than $1. Hint: this is really a question about the sum of the rounding errors. 
This is a homework problem and I have no idea where to even begin. Any help would be greatly appreciated. 
 A: The sum can be approximated by a nomrally distributed error. Its mean is obviously $0$, the variance is the sum of the variances of the single errors.
A: A random variable uniformly distributed on $\left(-\frac12,\frac12\right)$ has mean $0$ and variance $\frac1{12}$ 
so the sum of $32$ independent random variables with the same distribution has mean $0$ and variance $\frac{32}{12}=\frac83$. The value $1$ would then be $\sqrt{\frac38}$ standard deviations above the mean, and $-1$ the same below 
Using a normal approximation (justified using a central limit theorem argument) would suggest that the probability of the sum being in the interval $(-1,1)$ could be about $\Phi\left(\sqrt{\frac38}\right)-\Phi\left(-\sqrt{\frac38}\right)$ where $\Phi\left(x\right)$ is the cumulative distribution function of a standard normal random variable with mean $0$ and variance $1$.  So about $0.45971$
With a lot of effort involving numerical convolution, it is possible to do a more precise calculation, and I did so in 2008 in a note called May not sum to total due to rounding: the probability of rounding errors. On page 16 it gave a value for $G_{32}(1)$ of about $0.45804$
