Simplest is always “this is wakening #1”. 50% chance of escape, and soonest possible if it happens. Has the psychological disadvantage that if the first coin is tails, you’re stuck forever with no future chance of escape. You have no memory of any of them, so that is irrelevant—all that matters is the probability of escape—but it feels bad to us as outside observers.
You can stretch it out by randomizing your answer with a declining chance of higher numbers. Say, flip YOUR coin until you get tails, then guess the number of flips you’ve made. HHHT you guess 4, for example. This gives you 25% to be released on day 1 (50% DM’s coin is tails X 50% your first flip is.). And 6.5% to be released the second day (25% of HT on his coin and yours). Unfortunately, sum (i: 1->infinity) 0.25^i = 0.33333, so your overall chance of escaping is reduced. But you do always have some (very small) hope, unlike the simple answer.
Randomization with weighting toward earlier numbers improves your early chance, but reduces your later chances, and it seems (from sampling and thinking, not proven) that it can approach 0.5 but not exceed it.
I think the best you can do is 50% unless you have some information when you wake up about how long this has been going on.
Simplest is always “this is wakening #1”. 50% chance of escape, and soonest possible if it happens. Has the psychological disadvantage that if the first coin is tails, you’re stuck forever with no future chance of escape. You have no memory of any of them, so that is irrelevant—all that matters is the probability of escape—but it feels bad to us as outside observers.
You can stretch it out by randomizing your answer with a declining chance of higher numbers. Say, flip YOUR coin until you get tails, then guess the number of flips you’ve made. HHHT you guess 4, for example. This gives you 25% to be released on day 1 (50% DM’s coin is tails X 50% your first flip is.). And 6.5% to be released the second day (25% of HT on his coin and yours). Unfortunately, sum (i: 1->infinity) 0.25^i = 0.33333, so your overall chance of escaping is reduced. But you do always have some (very small) hope, unlike the simple answer.
Randomization with weighting toward earlier numbers improves your early chance, but reduces your later chances, and it seems (from sampling and thinking, not proven) that it can approach 0.5 but not exceed it.
I think the best you can do is 50% unless you have some information when you wake up about how long this has been going on.