Coin doesn’t help. Say I decide to pick 2 if it is heads, 1 if it is tails.
I’ve lowered my odds of escaping on try 1 to 1⁄4, which initially looks good, but the overall chance stays the same, since I get another 1⁄4 on the second round. If I do 2 flips, and use the 4 spread there to get 1, 2, 3, or 4, then I have an eight of a chance on each of rounds 1-4.
Similarly, if I raise the number of outcomes that point to one number, that round’s chance goes up , but the others decline, so my overall chance stays pegged to 1⁄2. (ie, if HH, HT, TH all make me say 1, then I have a 3⁄8 chance that round, but only a 1⁄8 of being awake on round 2 and getting TT).
Coin doesn’t help. Say I decide to pick 2 if it is heads, 1 if it is tails.
I’ve lowered my odds of escaping on try 1 to 1⁄4, which initially looks good, but the overall chance stays the same, since I get another 1⁄4 on the second round. If I do 2 flips, and use the 4 spread there to get 1, 2, 3, or 4, then I have an eight of a chance on each of rounds 1-4.
Similarly, if I raise the number of outcomes that point to one number, that round’s chance goes up , but the others decline, so my overall chance stays pegged to 1⁄2. (ie, if HH, HT, TH all make me say 1, then I have a 3⁄8 chance that round, but only a 1⁄8 of being awake on round 2 and getting TT).
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.