  A collection of graphs is called bridge-alterable if, for each graph G with a
bridge e, G is in the class if and only if G-e is. For example the class of
forests is bridge-alterable. For a random forest $F_n$ sampled uniformly from
the set of forests on vertex set {1,..,n}, a classical result of Renyi (1959)
shows that the probability that $F_n$ is connected is $e^{-1/2 +o(1)}$.
Recently Addario-Berry, McDiarmid and Reed (2012) and Kang and Panagiotou
(2013) independently proved that, given a bridge-alterable class, for a random
graph $R_n$ sampled uniformly from the graphs in the class on {1,..,n}, the
probability that $R_n$ is connected is at least $e^{-1/2 +o(1)}$. Here we give
a more straightforward proof, and obtain a stronger non-asymptotic form of this
result, which compares the probability to that for a random forest. We see that
the probability that $R_n$ is connected is at least the minimum over $\frac25 n
< t \leq n$ of the probability that $F_t$ is connected.
