The coin can at least lower your chances. Say, that you will say 3 if it is head and 4 if it is the tail.
You can win at round 3 with the probability 1⁄4 and you can win at round 4 with the probability 1⁄4.
Is that right?
Oh, yeah, I see what you are saying. Having 2 1⁄4 chances is, what, 7⁄16 of escape, so the coin does make it worse.
Sure. But not only to 7⁄16 but to the infinite number of other values, too. You just have to play with it longer.
The question now is, can the coin make it better, too? If not, why it can only make it worse?
If you say two numbers with nonzero probability, you can improve your chances by shifting all the probability mass to one of them.
The coin can at least lower your chances. Say, that you will say 3 if it is head and 4 if it is the tail.
You can win at round 3 with the probability 1⁄4 and you can win at round 4 with the probability 1⁄4.
Is that right?
Oh, yeah, I see what you are saying. Having 2 1⁄4 chances is, what, 7⁄16 of escape, so the coin does make it worse.
Sure. But not only to 7⁄16 but to the infinite number of other values, too. You just have to play with it longer.
The question now is, can the coin make it better, too? If not, why it can only make it worse?
If you say two numbers with nonzero probability, you can improve your chances by shifting all the probability mass to one of them.