For the probability that at least one arrow picked is cursed: 1/2 + 1/2 × 5/9 = 14/18 = 7/9. (The probability that you picked it the first time, plus the probability that you missed it the first time but picked it the second time).
Sidenote: it doesn’t matter whether you grab the arrows one at a time or with both hands at the same time. Unless your party is trying to figure out which arms to amputate.
For the probability that at least one arrow picked is cursed: 1/2 + 1/2 × 5/9 = 14/18 = 7/9. (The probability that you picked it the first time, plus the probability that you missed it the first time but picked it the second time).
a dice probability calculator tells me probability of 3d6 + 1d4 at least 16 is 2/9. So the DM is correct.