What are the odds of getting three of a kind in Three Card Poker?

This highly depends on exactly which Poker variant is being played.

In a simple draw of 5 cards from a deck there are 624 ways to get four of a kind. This is calculated by taking the 13 quads of each rank multiplied by the number of extra cards which can be the fifth card in the hand. There are a total of 2598960 5 card hands, so four of a kind has a chance of 624 / 2598960 = 0.024%

In Texas Hold'em there is a 0.15% chance for a player to get four of a kind. The chance is increased since there are now 7 cards rather than 5 from which to form a hand.