No. You will always say the same number each time, since you are identical each time.
As long as it isn’t that number, you are going another round. Eventually it gets to that number, whereupon you go free if you get the luck of the coin, or go back under if you miss it.
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).
Can you do worse than that?
Sure, you can guess zero or negative numbers or whatever.
Say, you must always give a positive number. Can you do worse than 1⁄2 then?
No. You will always say the same number each time, since you are identical each time.
As long as it isn’t that number, you are going another round. Eventually it gets to that number, whereupon you go free if you get the luck of the coin, or go back under if you miss it.
That’s why you get a fair coin. Like a program, which gets seed for its random number generator from the clock.
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.
If you say either 1 or 2 with probability 1⁄2 each, the probability of escaping is 7⁄16.
True. You can do it worse than 1⁄2. Just toss a coin and if it lands head up choose 1, otherwise choose 2.
You can link more numbers this way and it can be even worse.