  A class A of labelled graphs is bridge-addable if for all graphs G in A and
all vertices u and v in distinct connected components of G, the graph obtained
by adding an edge between u and u is also in A; the class A is monotone if for
all G in A and all subgraphs H of G, H is also in A. We show that for any
bridge-addable, monotone class A whose elements have vertex set 1,...,n, the
probability that a uniformly random element of A is connected is at least
(1-o_n(1)) e^{-1/2}, where o_n(1) tends to zero as n tends to infinity. This
establishes the special case of a conjecture of McDiarmid, Steger and Welsh
when the condition of monotonicity is added. This result has also been obtained
independently by Kang and Panagiotiou (2011).
