Suppose I have $k$ outcomes with probabilities, $p_i$, with $p_1+p_2+\dots+p_k=1$. Each probability has the same distribution. What would the distribution of a sum of probabilities be? For example, what would the distribution of $p_1+p_2$ be? The Irwin-Hall distribution is the distribution for the sum of uniformly distributed random variables. Each $p_i$ will be distributed between 0 and 1, but they won't be uniformly distributed. Does the Irwin-Hall distribution have anything to do with it?
I don't require a fully derived answer, but a kick in the right direction would help.