Request: If you were to roll a pair of unbiased dice (2d6) and total the result, until you rolled the same total 20 times, what are the chances of each value being the one you reach 20 on?

This came from a lesson I was in yesterday on experimental probability. We'd gotten the pupils/groups to roll 2d6 and record their result until they got 20 of the same number, then collated results and plotted them to see the distribution and compare it to the expected result. Most of the groups got fairly reasonable results, several finished with twenty 7's, but one boy managed to get to twenty 10's before anything else, and in fact did so in a remarkably low number of rolls (73 IIRC). The teacher and I were discussing afterwards about how unlikely this was and how you'd calculate it, and neither of us were especially confident in how to do it. I suspected it would be the hypergeometric distribution, but I've honestly barely used that so I'm not sure.

Any takers? If not I'll have to go over to Reddit's theydidthemaths.