Kyle's comment regarding expecting success during the End of the World 11 on 11's 3 out of 9 times got me thinking. So I did some research. I found the following article which provides information on 3rd down success rate by distance. I think these results are probably a close approximation of what we could expect in this scenario but are probably slightly inflated because a shortened field makes success less likely. Based on the article:
3rd and 5 is successful about 40% of the time. That would mean that with 3 independent efforts, you'd roughly have success 1.2 times.
3rd and 15 is successful about 15% of the time, which would be less than .5 times per 3 attempts.
3rd and 30 isn't listed but I'd wager doesn't work more than 8% of the time, or about .25 times per 3 attempts.
Overall success in those 3rd down situations would be between 1 and 2 times per 9 attempts. With the end zone restricting movement, I'd guess it closer to 1 than 2.
http://www.footballstudyhall.com/2013/5/7/4303984/college-football-third-down-percentages
3rd and 5 is successful about 40% of the time. That would mean that with 3 independent efforts, you'd roughly have success 1.2 times.
3rd and 15 is successful about 15% of the time, which would be less than .5 times per 3 attempts.
3rd and 30 isn't listed but I'd wager doesn't work more than 8% of the time, or about .25 times per 3 attempts.
Overall success in those 3rd down situations would be between 1 and 2 times per 9 attempts. With the end zone restricting movement, I'd guess it closer to 1 than 2.

http://www.footballstudyhall.com/2013/5/7/4303984/college-football-third-down-percentages