In the example with Bob, surely the odds of Bob having a crush on you after winking (2:1) should be higher than a random person winking at you (given as 10:1), as we already have reason to suspect that Bob is more likely to have a crush on you than some random person not part of the six.
Not if you consider that the 1:5 figure constrains that ONLY one person among the six has a crush on you. If you learn for a fact one does, you’ll also immediately know the others all don’t. Which is not true for a random selection of students—you could randomly pick six that all have a crush on you.
Bob belongs to a group in which you know for a fact five people DON’T have a crush on you. So you have evidence lowering Bob’s odds relative to a random winker.
Either that, or it doesn’t matter how many actually have a crush on you, you’re looking for the specific one you have definite evidence about. For thisy, a random winker is not qualified to enter the comparison at all—if they’re not one of the six, they’re not the person you’re looking for. So Bob might have a crush on you AND not be the person you’re looking for, although his odds are higher than those of the other five you don’t have any evidence about.
That’s the interpretations that make the math not wrong, anyway. If you only know that “at least one of them has a crush on me” and more than one could potentially satisfy your search criteria, the 1:5 figure is not the right odds.
What Liliet B said. Low priors will screw with you even after a “definitive” experiment. You might also want to take a look at this: https://www.lesswrong.com/posts/XTXWPQSEgoMkAupKt/an-intuitive-explanation-of-bayes-s-theorem