  This article advances the results of Duke on the average surjectivity of
Galois representations for elliptic curves to the context of Drinfeld modules
over function fields. Let $F$ be the rational function field over a finite
field. I establish that for Drinfeld modules of rank $r \geq 2$, the $T$-adic
Galois representation: $\widehat{\rho}_{\phi, T}: Gal(F^{sep}/F) \rightarrow
GL_r(\mathbb{F}_q[[T]])$ is surjective for a density $1$ set of such modules.
The proof utilizes Hilbert irreducibility (over function fields), Drinfeld's
uniformization theory and sieve methods.
