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

Question

This is part of my quest to untangle the controversy in The Snake Eyes Paradox. It's called Variant Y because it's how YES bettors interpreted 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 N 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 I either roll snake eyes and a group will be devoured by snakes, or I run out of people. Either way, I stop.

In the limit as N goes to infinity — an unlimited number of people — what is your chance of dying, given that you're chosen to play?

Argument for ~1/2

In the limit there's zero chance of running out of people which means a guarantee of a final group dying. That group 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 1/36

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.

In the limit, if I somehow never rolled snake eyes, then an infinite number of people would all play and none of them of would die. That has 0% probability but, y'know, hypothetically.

Resolution Criteria

This resolves to the mathematically correct probability, Pr(death | chosen). How do we decide what that is? I'm naively hopeful that this version, spelling out the limiting process up front, won't be controversial and we can resolve to consensus in the comments. If that's not realistic, I'll hide comments from everyone who disagrees with me and resolve to the consensus of what's left. Just kidding. Here's my new idea: a council of 2 people who've argued coherently for NO in the original market (eg, Martin Randall and Primer), 2 people who've argued coherently for YES (eg, David Pennock and Wamba Ivanhoe), plus a mathematician who hasn't bet (eg, Lorxus Mathfox). I predict we'll reach consensus but we'll vote if necessary. I won't trade in this market.