My estimated probability is extremely low because unlike the patient with anosognosia, I observe movement in my left arm. The patient did not observe motion and invoked absolute denial in inventing explanations for it.
Still, I'm not a diagnostician so I'll assume patients with anosognosia think they can actually observe motion in their left arms. In this case, it seems the popular proof is in doing something that requires both arms, like JulianMorrison did. Still, it seems that if you can deceive yourself about your arm moving, you can deceive yourself about such an experiment. It strikes me that one way to prove your left arm still works is to disable your right arm. After all, you have no illusions about your right side - if you disable your right arm, you'll know it. Therefore, I shall tape my right arm to the arm of the chair I'm sitting in, and finish typing this post with my left. There. It is currently 10:52 EST. The post timestamp will agree that this post was submitted before I released my right arm.
In retrospect, I advise against repeating this experiment, especially with duct tape, if you a) have hairy arms and b) have recently suffered a bad sunburn.
Followup to: The Strangest Thing An AI Could Tell You
Brain damage patients with anosognosia are incapable of considering, noticing, admitting, or realizing even after being argued with, that their left arm, left leg, or left side of the body, is paralyzed. Again I'll quote Yvain's summary:
A brief search didn't turn up a base-rate frequency in the population for left-arm paralysis with anosognosia, but let's say the base rate is 1 in 10,000,000 individuals (so around 670 individuals worldwide).
Supposing this to be the prior, what is your estimated probability that your left arm is currently paralyzed?
Added: This interests me because it seems to be a special case of the same general issue discussed in The Modesty Argument and Robin's reply Sleepy Fools - when pathological minds roughly similar to yours update based on fabricated evidence to conclude they are not pathological, under what circumstances can you update on different-seeming evidence to conclude that you are not pathological?