Hornblower's Duel

[Lin Sherman]

Horatio Hornblower is accused of cheating at cards, and to preserve his honor, challenges his accuser to a duel. As the challenger, Hornblower gets to pick the weapons. His opponent is an expert at every sort of weaponry, while Hornblower is a bookish nerd, so Hornblower comes up with the following scheme to maximize his chances of survival: he decides the duel will be conducted with pistols, both of which will be loaded with powder, but only one with shot. This means his opponent, an expert marksman, will have a .5 probability of selecting the lethal pistol.

The pistols of the late 18th century, however, were wildly inaccurate, and even in an expert marksman using a highest quality pistol would have only a .8 probability of hitting his target. So Hornblower's actual probability of being killed is only .5 x .8 = .4

As it turns out, Hornblower (who throws down his pistol without taking a shot at his opponent) lives. What is the probability that Hornblower's opponent selected the pistol loaded with shot?

Lin