How likely is it that the Sun will destroy our electric society? Given the studies of the Sun and similar stars and the most impressive understanding of its physics, is there any kind of probability distribution for its dangerous behavior? There are also some historic data about events in 1859 and 775. What is the estimated probability of solar activity destroying our electric grid within this century?
I should give credit here to @honeste_vivere, who pointed out to me today that there have been recent studies excited by some extremely large coronal mass ejections that were classed as "near misses" in terms of causing major disruption.

Of particular interest to you would be the event of July 2012 discussed by Baker et al. (2013). I quote from the paper

" Had the season and time of day for this CME passage been right on striking the Earth, the world would have witnessed a storm larger (possibly much larger) than the 1859 Carrington event. This most likely would have had devastating consequences for many technological systems "

and then in the discussion

"
It is the opinion of the authors that our advanced technological society was very fortunate, indeed, that the 23 July solar storm did not occur just a week or so earlier.
Had the storm occurred in mid-July 2012, the Earth would have been directly targeted by the CME and an unprecedentedly large space weather event would have resulted. In fact, there is very legitimate question of whether our society would still be “picking up the pieces” from such as severe event [see NRC, 2008].
"

It looks likely that such  events are perhaps things that occur every few decades. Riley et al. (2012) suggest something as powerful as the Carrington event has a 12% chance of occurring in the next decade, with a 1% chance of something several times bigger. Even more powerful events are seen on other stars, where both flare and CME energies follow a power law relationship of the form $dN/dE \propto E^{-\alpha}$, where $\alpha \sim 2.5$ (Drake et al. 2013). Thus much larger, but rarer flares are possible and also seen on (usually) much younger and faster rotating stars. There may however be a very significant sporadic tail to the solar flare energy distribution too. Recent observations of candidate "superflares" on otherwise unremarkable solar-type stars have been reported, though the reality of these and the mechanisms are still being explored. Shibayama et al. (2013) There appear to be a group, consisting of 1% of stars that show (repeated) superflares more than 100 times the size of the Carrington event. These appear a little less likely around stars rotating as slow as the sun, but the bottom line is that, if there is nothing "special" about these stars, then these events occur every 800-5000 years (for events of energy 1e34 - 1e35 ergs) on a slow-rotating G-star. (Though you need to read carefully - it is probably a function of temperature and rotation). The authors note that superflares may be associated with the formation of very large starspots (or starspot groups). So that may be our clue to duck and cover.