In the English Vocabulary Profile, the adverbial phrase ‘at random’ is listed at C1 with the meaning:
chosen by chance
For example:
So we‘ll choose one at random then.
A collocate search in the iWeb corpus:
1 CHOSEN 4713
2 SELECTED 4295
3 TIMES 2328
4 DRAWN 1694
5 PICKED 1493
6 PICK 1155
7 WINNER 1068
8 ENTRIES 999
9 INTERVALS 928
10 WINNERS 707