First, pretend that the Mathematicians know there are an even number of white hats, and an even number of black hats. In that case it's easy for each one of them to guess what color hat they're wearing. So have the Mathematicians assume there are an even number of each type of hat. There's a 50% chance that the number of hats actually is even (non-trivial, but I'll leave out details for now), and therefore a 50% overall chance that the Mathematicians get every guess correct (and a 50% chance they get every guess wrong!)
(Replying to PARENT post)
First, pretend that the Mathematicians know there are an even number of white hats, and an even number of black hats. In that case it's easy for each one of them to guess what color hat they're wearing. So have the Mathematicians assume there are an even number of each type of hat. There's a 50% chance that the number of hats actually is even (non-trivial, but I'll leave out details for now), and therefore a 50% overall chance that the Mathematicians get every guess correct (and a 50% chance they get every guess wrong!)