OK I give up. I started 3 different solutions and got lost in the reasoning in all 3 of them.
I even began again and tried to go super simple,
1-- (on the day before 100th prisoner)
B sees 53 B & 45 G = 98 prisoners
2-- (prisoner #100 comes in with a B mark)
B now sees 54 B and 45 G = 99
but went blank on where to go from there.
Maybe, I'm not very logical?