Establish that
$$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = -1\right\}$$
and deduce the probability that a permutation of $\mathfrak{S}_{n}$ drawn uniformly at random has a prescribed signature.