In Ezra 2:62-63, the people who couldn't prove their descent from Israel need to wait for a priest to determine their true ancestry using the Urim and Thummim. These were (apparently) stones used to perceive the truth clairvoyantly—a form of divination sanctioned by God. They were originally worn on the breastplate of the high priest back in the days of the Tabernacle, a kind of mobile temple, after the Exodus from Egypt.
So here the Israelites are relying on an ancient method (or a version of it) to try to make these decisions, connecting themselves with their traditions and practices at the same time that they're trying to determine whether they should or can let these people in.
Fun fact: the words urim and thummim in Hebrew are part of the crest of Yale University. Evidently, the admissions committee uses this same technique for predicting who won't make it through organic chemistry even though they claim to want to be pre-med.