Ignoring the prior probability invalidates the argument, but it cuts both ways. Consider a drug sniffing dog at an airport, it sniffs all carry-on bags as peop

Ignoring the prior probability invalidates the argument, but it cuts both ways. Consider a drug sniffing dog at an airport, it sniffs all carry-on bags as people go through the line to be searched. If we find drugs even 10% of the time that the dog whines, it's amazing, because the actual distribution of drugs amongst travelers is minuscule, maybe 0.01 of 1%. But if the dog is right 55% of the time? This is excellent probable cause for a search. But by way of contrast, consider a drug-sniffing dog that is only brought in when a human handler suspects a traveller may be carrying drugs. If the handler is very good at their job, and optimizing for success, the handler may themselves have a 55% positive-positive rate. Meaning, 55% of the people they pick are carrying drugs. Now if the dog whines 55% of the time they are given a suspect that has a 55% chance of carrying drugs, well, that is no better than a coin flip, it's garbage. And of course, what if the handler is amazing at picking people who might be carrying drugs? For example, what if they have access to warrentless surveillance meta-data and are using the dog as an excuse for parallel reconstruction ("Your honour, I picked this person at random, and the dog whined, so I searched them"). The handler might have a near 100% success rate of picking people carrying drugs. Now a 55% rate of whining is truly terrible for deciding whether the dog is any good.