At this point you've had this explained to you multiple times. May I suggest that if you don't get it at this point, maybe be a bit of an empiricist and write a computer program to repeat the game many times and see what fraction switching wins? Or if you don't have the skill to do that (in which case learning to program should be on your list of things to learn how to do. It is very helpful and forces certain forms of careful thinking) play the game out with a friend in real life.