Let's say that the probability of winning a regional is 3/50, and we want to win at least one regional. Well, then the probability of NOT winning a regional is 47/50. THIS quantity will multiply.
So then we say, well, the probability of winning at least regional as a function of N, where N is the number of regionals we attend, is:
P(N) = 1 - (47/50)^N
Because (47/50)^N is the probability of NOT winning a regional, and the sum of the probabilities of all outcomes must equal 1.
And the result?
1 Regional: 3/50 or 6%
2 Regionals: 11.64%
3 Regionals: 16.95%
Note that this is the probability of winning at least one regional. If you wanted to get the probability of winning exactly two, then it would be (3/50)*(3/50)*(47/50). The probability of winning three would be (3/50)*(3/50)*(3/50). So the teams that won three had a 0.022% chance of doing so. Congratulations, 1114 and 1024, you guy beat the odds!
Or maybe winning multiple regionals has more to do with robot quality, drive team skill, and a good autonomous than pure luck...though luck certainly is always involved.
Thus, it does go up (given that EVERYTHING else is equal, which it isn't) which makes sense--more chances equals more probability, but it doesn't quite scale linearly with regionals. Think about it, if they just added, then if you had a 3/50 shot of winning a regional, then if you went to 17 regionals you would be guaranteed a berth at Nationals, and your probability would be OVER 1.0, which is not really possible.