I don't think they placed bets, but Acts is pretty clear that after they nominated two Joseph and Matthias, they prayed and they cast their lots and the lot fell on Matthias.
I think it is very interesting to look at this election method and compare it to modern methods. We Anglicans elect our bishops. While the exact method varies from one diocese to another, it boils down to majority rules. I don't know whether that's the best way to do it. I suppose in a strict sense it's not the biblical way, but I see it as an administrative issue rather than a docrinal one.