Hey yvonnehr.

Recall that the sum of normal distributions is a normal distribution where the final variance is the sum of all individual variances (if they are all independent) and the mean is the sum of all the means.

If you have n independent observations with variance sigma^2 and mean mu then the sum will be distributed Normal(n*mu,n*sigma^2) which means you can use this to calculate a probability of the sum of the weights.