Is the probability of dying in the Snake Eyes Paradox 1/36?

Question

You're offered a gamble where a pair of six-sided dice are rolled and unless they come up snake eyes you get a bajillion dollars. If they do come up snake eyes, you're devoured by snakes.

So far it sounds like you have a 1/36 chance of dying, right?

Now the twist. First, I gather up an unlimited number of people willing to play the game. I take 1 person from that pool and let them play. Then I take 2 people and have them play together, where they share a dice roll and either get the bajillion dollars each or both get devoured. Then I do the same with 4 people, and then 8, 16, and so on.

At some point one of those groups will be devoured by snakes and then I stop.

[Alt. phrasing: We keep going until one of those groups is devoured by snakes, then the game stops.]

Is the probability that you'll die, given that you're chosen to play, still 1/36?

Argument for NO aka the frequency argument

Due to the doubling, the final group of people that die is slightly bigger than all the surviving groups put together. So if you're chosen to play you have about a 50% chance of dying! 😬 🐍

Argument for YES aka the one-fair-roll argument

The dice rolls are independent and whenever you're chosen, whatever happened in earlier rounds is irrelevant. Your chances of death are the chances of snake eyes on your round: 1/36. 😅

Clarifications and FAQ

The game is not adversarial; the dice rolls are independent and fair.

Choosing each group also happens uniformly randomly and without replacement.

[Retracted: An attempt to define a truncated game to take a limit of.]

[Retracted: Blah blah conditional probability.]

[Retracted: Whether it's possible for everyone to survive.]

[Retracted: More blah-blah on conditional probability.]

[Retracted: Ruling out "undefined" as an answer.]

[Retracted: Nitpicking "at some point".]

[Retracted: Yet more blah-blah on conditional probability.]

Resolution criteria

For an official resolution we'll write up a proof (or "proof") that the answer is 1/36 and a proof that the answer is ~1/2 (really anything greater than 1/36 would be fine) and then recruit some mathematician(s) to make an independent judgment on which is correct. Or maybe we'll just reach consensus in the comments?

UPDATE: @Lorxus has accepted the role of adjudicator. Here are Lorxus's terms, with some comments of mine in brackets:

0. [For a fee of 1% of market volume at the time of acceptance, M$1308, now paid], I will continue to uphold my commitment not to bet on this market. More importantly, now that @dreev has passed arbitration to me, I will read the mathematical arguments of both sides and use my best professional judgement to diligently determine soundness and mathematical correctness.

1. The YES bettors and the NO bettors must reach their separate consensuses as to which writeup to send me. I will only accept a single writeup per side, and will only accept these writeups in the form of a PDF or text file. If you choose to use LaTeX, send me the PDF and not the raw TeX. This must be sent to me before 2023-09-30 at 23:59 EST. I will treat any failures in this regard as a blank entry.

2. The YES bettors and the NO bettors must each choose at most 3 champions per side. The role of these champions will be threefold: to contact me to send me the official writeup of their side, to answer reasonable mathematical questions about the writeups that I pose to them, and to pick formal holes in the other side's writeup to present to me. These champions must be selected and the outcome communicated to me before 2023-09-30 at 23:59 EST. I will treat any failures in this regard as a forfeit.

3. All bettors of this market agree that my judgement will be final, binding, and at my sole discretion. [And other terms about harassing Lorxus but I, @dreev, am insuring against such and believe we don't need to worry about this. Everyone just be nice, ok??]

4. If only one such writeup is sound and mathematically correct, I will choose it as the winner. If only one side's champions can find flaws in the writeup of the other's, or only one side can resolve any flaws I point out in their writeup, I will choose it as the winner. If somehow neither writeup is sound and mathematically correct, and this difficulty persists equally for both sides through questioning, I will say so, and the market will resolve N/A, making everyone including me unhappy. If both writeups are approximately sound and mathematically correct, then I will move to cross-questioning, and I will determine the winner by subjective severity of the worst irresolvable flaw any champion can pick in either writeup. If at that point I have reached the conclusion that the question was fundamentally and irresolvably ill-posed, or both sides have found comparably good flaws, I will arbitrate that the market be resolved to 30-day average of market price. [Resolve-to-MKT is fraught and we may need to hash this out better to ensure fairness -- @Lorxus mentions "anti-spiked" MKT...

Manifold Markets · Accepted Answer

Yes — resolved on Nov 2, 2023 by Manifold Markets prediction market.