I overthought it and made a grid for number crunching.
0.2 0.2 0 0.6
0.282759 0.282759 0.0827586 0.351724
0.297537 0.297537 0.203941 0.200985
0.276081 0.276081 0.33618 0.111658
0.238137 0.238137 0.463602 0.0601238
0.195414 0.195414 0.577907 0.0312644
0.154377 0.154377 0.675615 0.0156322
0.118182 0.118182 0.756159 0.00747626
First and second columns are identical, they're the chances of getting one or the other. Second is the chances of getting both after the draw. Last column's getting none.
Auto-mil works at the seventh draw, but the chances of it happening are 1.56%, which is pretty insignificant in comparison. So, the best estimation of getting it on the first turn is about 68%, while you'll have about a 76% if you're going second.
Still fairly good odds, though.