See: /dreev/is-the-probability-of-dying-in-the

/dreev/whats-the-probability-of-death-in-s-770398272e8d

The Snake Eyes paradox involves a game where the GM selects people from an infinite pool, starting with 1 person on Round 1. On each round, he rolls 2 fair six-sided dice. If they land snake eyes (which has a 1/36 chance of happening, assuming the dice are fair), the game ends and the people chosen for that round die. Otherwise, the game goes to another round in which twice as many people are chosen.

The original paradox asks what your credence should be that you will die, given that you have just been chosen for the game. The answer to this question is definitely 1/36 - it's unambiguous in both classical probability theory and a non-standard variation of probability theory.

Variant N asks what your credence that you will die should be, given that you know the game eventually ends (it's otherwise possible for it to go on forever if snake eyes is never rolled). In classical probability theory, this is still 1/36, since the probability of never rolling snake eyes was zero anyway. However, classical probability theory doesn't allow for a uniform distribution over the natural numbers, which means you must assume that you are more likely to have been chosen for the game than all but finitely many other people, no matter what your priors are. It is possible to instead reason in a non-standard theory of probability, in which the GM has an equal probability of picking any of the remaining people in the infinite pool every time he needs to choose a new player. In this variant, it turns out that, even though the prior probability of an infinite game is 0, the probability of an infinite game conditional on you being chosen is 17/18, and thus, the probability that you die, given that the game is finite, is 1/2.

NNN stands for Non-standard, Nth round, Variant N.

Thus, Variant NNN asks what the probability that you die should be in the non-standard probability theory mentioned above, given that you are chosen on the Nth round of the game, and that you know the game will eventually end.

Argument for 1/36

The dice are fair, so they have a 1/36 chance of landing snake eyes on Round N and every round thereafter, and this probability is independent of whoever was chosen for Round N. Furthermore, your being chosen on Round N is not made more or less likely by the dice landing in any particular way on Round N or later rounds. Therefore, your probability for any event E that only depends on the roll of the dice after you are chosen on Round N, conditional on you being chosen for Round N, should be the same as your probability that E occurs conditional on Round N occurring at all (since you being chosen, rather than anyone else, was independent of this). Thus, once you learn that you were chosen on Round N, you should update your probability that the game goes to infinitely many rounds back to 0, since there is still a 0% chance that the game goes on forever, given that it goes to at least Round N. You should also believe that you have a 1/36 chance of dying in this case, just like regular snake eyes, since your chance of dying is independent of the Round you are chosen on.

Given this, learning that the game eventually ends gives you no additional information to update your credence. Specifically, before this, you had P(D|C_N) = 1/36, where D means "You die," and C_N means, "You're chosen for the Nth round" and P(F|C_N) = 1, where F means a finite game. So learning F causes no update to your credence:

P(D|C_N,F) = P(D|C_N)P(F|D,C_N) / P(F|C_N) = P(D|C_N) = 1/36.

Argument for ≥1/2

According to non-standard variant N (but without the condition that you know which round N you were chosen on), P(D|C,F) = 1/2, where C means that you're chosen to play the game on any round. Therefore, if P(D|C_N,F) is independent of the round number N, it must be exactly 1/2, since any other value A would mean that, in the situation where you know that you were chosen and the game was finite, learning what your round number N was would force you to update your credence from 1/2 to A no matter what N is. This is absurd: If you already know that you should update your belief to A no matter what N is, then you should just update it now. Therefore A could only be 1/2, so the argument for 1/36 must be flawed.

In reality, P(D|C_N,F) doesn't have to be exactly 1/2 for all N. It just needs to have expected value of 1/2, in line with the law of total expectation. Because of the unusual characteristics of the distribution, this just requires the limit of P(D|C_N,F) as N approaches infinity to be 1/2. For individual N values, the number may be slightly higher, since it would be related to the expected proportion of people who die. (More details on the non-standard distribution and Variant N can be found in my Substack post).

Argument for undefined

Non-standard probability theory is non-standard for a reason. For some questions, it cannot produce a coherent answer. The two previous arguments are both valid and show that no probability can be consistently defined for this question.

Resolution Criteria

I will resolve this once I am confident that there is a correct answer, and I know what it is, to whatever the correct probability is. The close date will be extended until this happens. If I become convinced that the problem itself is meaningless or self-contradictory, or that the correct answer is that there is no well-defined probability, I will resolve to "Undefined". If I become convinced that the best answer involves infinitesimals, I will resolve to the finite part of the answer (e.g., "1/36+infinitesimal" still counts as 1/36 for this market). If the problem as stated is ambiguous such that there are two equally plausible interpretations, I will resolve 50-50 to both interpretations if they have different answers. I will only do this if both interpretations are equally plausible - if one is more plausible than the other or definitely more in line with what I was asking, I will resolve purely based on the more plausible interpretation.

Whatever answer I resolve it to must be true for all values of N. So, for example, if I thought the answer was 1/36 for N=1 but 1/2 for N=2, I would not resolve it to either 1/36 or ≥1/2 - both options would resolve to NO. If none of the options are correct for all values of N, I will resolve to Other.