In the "Probability" section, you say:

Suppose you start out 85% confident that the one remaining enemy soldier is not a sniper. That leaves only 15% credence to the hypothesis that he is a sniper.

But in the next section, "The Problem of Priors", you say:

In the example above where you're a soldier in combat, I gave you your starting probabilities: 85% confidence that the enemy soldier was a sniper, and 15% confidence he was not.

Seems like you swapped the numbers.