The maximum degree of a random graph

Let 0 < p < 1, q = 1 - p and b be fixed and let G epsilon G(n,p) be a graph on n vertices where each pair of vertices is joined independently with probability p. We show that the probability that every vertex of G has degree at most pn + b root npq is equal to (c + o(1))(n), for c = c(b) the root of a certain equation. Surprisingly, c(0) = 0.6102... is greater than (1)/(2) and c(b) is independent of p. To obtain these results we consider the complete graph on n vertices with weights on the edges. Taking these weights as independent normal N(p,pq) random variables gives a 'continuous' approximation to G(n, p) whose degrees are much easier to analyse.