Black's best option is a non-violent one: to shoot into the air.
If he shoots at Gray and kills him then he is a dead man. If he shoots at White and kills him then he only has a 1/3 chance of survival. By shooting into the air he ensures that Gray and White shoot it out and then he has the first shot against the survivor. In other words, by initially killing one of his opponents he would only make his chances worse because then the remaining opponent would shoot at him instead of at the third man.
If Black shoots into the air the survival probabilities are:
White 22.22 %
Gray 38.09 %
Black 39.69 %
It seems ironic that the worst shot has the best chances, and that this is because he is the least dangerous. Sounds like the tall puppy syndrome.
The above solution - shooting into the air - is that given in Singh's book. Someone on the Net alerted me to a further flourish, namely that there is nothing to stop Gray and White from thinking along the same lines. In fact, once Black shoots into the air, they are likely to do the same thing, if they have any sense. However, this does not invalidate the solution given.
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