That can be done much more securely with a cryptographic coin flip. Have each applicant generate some random numbers (say 16 bytes worth) and then publicly release the MD5 (or similar post quantum proof crypto hash) of the numbers.

After an agreed-on period of time, all applicants publicly release their original random numbers. They're all hashed together and the result is used for the random hat drawing. If an applicant refuses to / can't release their original data, they get struck from the pool and this is run again. This should rarely happen.

This procedure may need tweaked a bit (IANACryptanalyst), but it's called a distributed coin flip algorithm. The nice thing about this vs. AI is that it is fair even if all but one participant is trying to rig it. With AI - you have a centralized system - far easier to cheat. Someone finds some type of injection attack on the AI and democracy's over.