It's especially confusing when the other use that you have for probability of evidence given hypothesis is not mentioned.

Suppose you have a non-sniper test which has 10% probability of returning nonsniper if enemy is a sniper, and near-certainty of returning nonsniper if enemy is not a sniper. Lacking a prior, you may not know how likely it is that it is a sniper, but you know that executing a program do the test 3 times then run if all indicate non-sniper has expected risk of running out into sniper of 0.1% or less if the tests are statistically independent. This is very useful when determining advance policies such as the testing required for some airplane hardware component, or for a drug. Or when designing an AI for a computer game, and in many other competitive situations.