Two machines, A and B, both have a light bulb on them. For machine A, the light flashes red with .8 prob and blue with .2 prob. For machine B, .2 prob red and .8 prob blue. With .5 probability you are presented with machine A or B. Your task is to observe the color of its flashes, so as to make a best guess whether it's A or B.
Before you start to observe, however, you must make a decision as to how many times you want to observe its flash (say n times), and after n times, you make your guess.
Intuitively I thought the larger n is, the better our chances are. But when I calculated n=2, I found that you still have a .8 prob of being correct, the same as when you observe only once. Is it true that no matter how many times you observe, you can't do better than .8? I find it baffling.