Is competing at multiple regionals REALLY fair ?

[Nikhil Bajaj]

Originally Posted by EricH

We assume each team has an equal chance of winning one of the 4 non-submission, non-rookie awards. That is not the case, but we want a best-case scenario here.

I think your logic may also be flawed. You're saying that in two events, a team has a 6/50 chance. They have a 3/50 in each (using the numbers we're playing with). However, as I remember probability, you don't add. The odds of x event in two separate cases is multiplied. Your chances of qualifying twice are much smaller than indicated.

Assuming, of course, that everything is equal, which it isn't.

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.