What's the probability of death in Snake Eyes Variant N?

Question

This is part of my quest to untangle the controversy in The Snake Eyes Paradox. It's called Variant N because it's how NO bettors intepreted the original.

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 infinite 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.

When snake eyes is eventually rolled, that's the final group. They're devoured by snakes and we're done.

[EDIT and META-EDIT: I had added a thing here about always rolling snake eyes "even in the finite version of the game" but part of the point of Variant N is that there's no finite/truncated version. There's simply an infinite pool of people.]

What is your chance of dying, given that you're chosen to play and that snake eyes gets rolled in a finite number of rolls?

Argument for ~1/2 aka the frequency argument

We're explicitly guaranteed that we'll roll snake eyes and when we do, that's a final group dying that's 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 1/36 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 and the dice rolls are independent and truly random.

Choosing each group also happens uniformly randomly and without replacement.

What if the question turns out to be self-contradictory or the answer isn't a real number? Then we argue about how to fix it in the comments and do our best to agree on the best possible real number answer.

Resolution Criteria

This resolves to the mathematically correct probability, Pr(death | chosen). How do we decide what that is? With a council of 2 people who've argued coherently for NO in the original market, 2 people who've argued coherently for YES, plus a mathematician who hasn't bet. See the companion market, Snake Eyes Variant Y, for details. I won't trade in this market.

Manifold Markets · Answer

Roughly even odds — Manifold Markets prediction market estimates a 48% chance (9 traders, as of Jul 4, 2024).