There is a class of probability puzzles that includes Monty Hall/Three Prisoners, Three Cards/Pancakes, Two Children/Boy or Girl, their common antecedent Bertrand's Box Paradox, and (a more complicated variation) Tuesday Boy. Any one of them will generate controversy, even among so-called experts. What I find interesting, is that some experts (including in publication: Devlin, Mlodinow, self-proclaimed expert vos Savant, and even Martin Gardner before he changed his answer) take what I consider to be both sides for different puzzles in this list.
My question is not "what IS the answer." I know the answers to all, but expect arguments for some of them. I ask for opinions on why the controversies exist (I also think I know this answer, but will defer stating it).
The common theme is that one of N objects is selected at random. At least one of two functionally-equivalent facts (call them B and G) apply to each object. Both B and G apply to M objects, but only one fact applies to the other (N-M), evenly divided between B and G. If you know, through unspecified means, that B applies to the selected object, what are the chances G also applies? (Note: the problem more frequently asks for the reverse probability, that G doesn't apply.)
The two debated answers are M/N and 2M/(N+M). What is the difference, and why should one choose one over the other?
I will select Monty Hall problem. Simplest view - for a player who is playing game it will be really confusing. He'll search a lot on internet, math books and ask advice to PhD holder's.
But if host chooses different player who didn't know what first player had choose. (of-coarse after revealing an empty door). It will be 50-50 chance for him. Either win it or lose it.