Please first have a look at the following little problem:
There are two indistinguishable light bulbs A and B. A flashes red light with prob .8 and blue with prob .2; B red with .2 and blue .8. Now with .5 prob you are presented with either A or B. You're supposed to observe its flash color to make a best guess (maximizing probability of correct guessing) which bulb it is. Before you start to make observations, however, you must decide how many times you want to observe it (say n times, then you observe it flashing n times and make your guess). Suppose flashes are independent.
Intuitively, one would think the more observations one makes, the better one's chances will be. Curiously though, it is easy calculation to show that n=2 doesn't improve upon n=1, and n=4 doesn't improve upon n=3. I didn't go further but I speculate n=2k doesn't improve upon n=2k-1. I'm not able to prove it for the general case. But is it true? If so, how can one intuitively understand the result?