Ah, I get it now. N is stated ahead of time, but you can (predictably) use the information so far to “stop playing” whenever you want, and that’s part of the optimal strategy to consider in the expectation.
Yeah, which means if I’m trying to maximize my payout, I’ll set N arbitrarily large and abort the game at sufficient evidence that the coin isn’t predictable enough for the game to have positive expected value. If the coin is predictable enough, then I’ll pump my friend for every last cent he has.
However, note that the problem as stated asks for the minimum value of N so that the game has positive expected value. (I’m not too sure why we’re interested in this except as an exercise).
edit: just clarifying for others. Not that I think you misunderstood.
So then can N be the dynamic result of a function, rather than a value?
No, you have to state N before you start flipping coins.
Ah, I get it now. N is stated ahead of time, but you can (predictably) use the information so far to “stop playing” whenever you want, and that’s part of the optimal strategy to consider in the expectation.
Yeah, which means if I’m trying to maximize my payout, I’ll set N arbitrarily large and abort the game at sufficient evidence that the coin isn’t predictable enough for the game to have positive expected value. If the coin is predictable enough, then I’ll pump my friend for every last cent he has.
However, note that the problem as stated asks for the minimum value of N so that the game has positive expected value. (I’m not too sure why we’re interested in this except as an exercise).
edit: just clarifying for others. Not that I think you misunderstood.