Were any tests done, either back then or in the modern day?
Also, is it possible that the user of the pistols could influence the rate of misfires? For instance, if loading the pistol improperly could cause a misfire, then we might reasonably expect both pistols to be loaded improperly, and a double misfire might be more likely.
Well, the pistol wasn't a flintlock. The February 2 1835 National Gazette stated that the "percussion cap exploded with a noise so great that several witnesses supposed the pistol had fired". So we can rule out flintlock problems: improperly sharpened flint, damp powder in the pan, plugged touch hole. We know that the cap went off. There are statements online about the pistols being tested and fired with no trouble afterwards. Assuming that the pistols were mechanically OK, dampness is the most likely explanation, and the common hypothesis seems to be that damp weather had something to do with it. Black powder is notoriously hygroscopic and , when damp, resists exploding- water vaporization effectively removes enough heat to prevent the temperature from rising enough to start the reaction of the potassium nitrate and sulfur with the carbon. Since a muzzleloading pistol is loaded with loose powder, there would have been many ways for the powder to become damp if the day was rainy or misty. And if the two pistols were loaded in the same way at the same time and place, the powder could have been equally damp in both. Lawrence may not have been careful about loading: he was found not guilty by virtue of insanity, after all.
With a quick search I can't find any period sources online for the trials of the pistols afterwards : the best is likely an 1835 pamphlet from a Washington reporter sitting over at the Library of Congress. It would be very interesting to know whether that meant they had simply put new caps on the pistols and tried them again, or pulled the charges and looked at the powder, then reloaded with it...which might have allowed it to dry enough to work. It might be that Lawrence would have been more successful if he had simply put a new cap on the first pistol and tried another shot, instead of pulling the second pistol.
Just to add to /u/Bodark43's comment:
I am assuming (since OP didn't provide a source) that the 1-in-100,000 odds would represent the odds of two consecutive misfires assuming the events were completely independent of each other.
Which almost certainly wasn't the case, as the other comment makes clear: if the guns were loaded at the same time, with the same powder, same everything, then the odds of the second gun's misfiring given that the first had already done so are actually quite high.
It's a classic statistical error (whether intentional or not) to calculate the odds of Events A and B assuming they're independent of one another when they're not.