Anyway, towards the end of the paper things are put into the historical context of the theory of group extensions, and we learn that:
The groups that can be obtained from the trivial group by iteratively building cyclic extensions are called solvable. Statistically speaking “most” groups are solvable. For instance, all groups of order less than sixty are solvable.Of course, groups of order less than sixty make up a negligible portion of all groups; it's certainly possible that all groups of order less than sixty are solvable, but most groups of order n are not solvable when n is large enough. There is the Feit-Thompson theorem, which states that every group of odd order is solvable, but numerical data for n up to 2015 shows that there seem to be more groups of even order than odd order. This isn't surprising; the number of groups of a given finite order depends strongly on the exponents of the primes in its factorization. Nor is this special to 2 -- in general numbers with lots of prime factors are the order of lots of groups.
It looks like the statement is true, though, in any reasonable sense; from Sloane's Encyclopedia, there are only 2240 integers n below 105 such that there exists a non-solvable group of order n. In fact, it looks like the sequence of such numbers has roughly constant density, of about 0.0224. That should be provable or disprovable from the note given in the Encyclopedia, which gives an easy way to find such numbers. The statement that most numbers are solvable (where a number is solvable if all groups of that order are solvable) is even stronger than the statement that most groups are solvable.
But talking about a ``randomly chosen group'' seems a bit silly, anyway; the groups of a given order don't strike me as the sort of set one picks things from at random. Choosing a random element of a group seems like a much more natural operation. (Feel free to correct me if I'm wrong!)