Puzzle of the Week #123 - Forensic Probability

Imagine two games:

In one a person rolls a dice repeatedly and notes down the result. If a particular number appears three times, any subsequent roll of that number will be ignored. This is continued until ten numbers have been written down.

In another there is a pack of 18 cards, three each with the numbers 1 to 6 on them. 10 cards are drawn at random. The ten numbers are written down.

In both cases the result will be ten numbers between 1 and 6, with no number appearing more than three times. It will be impossible to know whether the ten numbers came from the dice or the cards.

Is it possible to detect, given the results of many repeated trials, which game someone is playing, or are the two games essentially identical? (Iā€™m not for numbers, just general trends that one might look out for).