A Bayesian Belief Network (BN) is a model of a joint distribution over a finite set of variables, with a DAG structure to represent the immediate dependencies between the variables, and a set of parameters (aka CPTables) to represent the local conditional probabilities of a node, given each assignment to its parents. In many situations, the parameters are themselves treated as random variables --- reflecting the uncertainty remaining after drawing on knowledge of domain experts and/or observing data generated by the network. A distribution over the CPtable parameters induces a distribution for the response the BN will return to any ``What is P(H | E)?'' query. This paper investigates the distribution of this response, shows that it is asymptotically normal, and derives closed-form expressions for its mean and asymptotic variance. We show that this computation has the same complexity as simply computing the (mean value of the) response --- ie, O(n * exp(w)), where n is the number of variables and w is the effective tree width. We also provide empirical evidence showing that the error-bars computed from our estimates are fairly accurate in practice, over a wide range of belief net structures and queries.