Is the probability of "death" in the ball-and-urn version of the Snake Eyes Paradox 1/36?

Question

The snake eyes market has generated a lot of debate (1.1K comments and counting!). One aspect of the debate is the question of anthropics. So we have an explicitly anthropic version of snake eyes.

Here's a version that attempts to factor out the question of anthropics altogether:

We have an urn with n balls. We choose 1 ball randomly and without replacement, roll some dice, and deflate it if we roll snake eyes. If we don't roll snake eyes, we repeat with 2 balls, then 4, then 8, and so on. We stop when we roll snake eyes and deflate the last batch of balls, or the urn is empty, whichever happens first.

Now randomly choose a ball from outside the urn.

In the limit as n goes to infinity -- i.e., an urn with an unlimited number of balls -- what is the probability that that ball is deflated?

Argument for NO: Assuming we don't run out of balls -- which in the limit we won't -- slightly more than half of those removed from the urn end up deflated, due to the doubling batches. So a randomly chosen one has a ~50% chance of being deflated.

Argument for YES: Put yourself in the shoes of one of the balls. The setup allows for exactly one way for you to be deflated: as the result of a single fair dice roll.

Resolution criteria

I'll defer to Martin Randall's opinion unless I can articulate why I'm certain he's wrong. Since this could be subjective, I won't trade in this market.

Clarifications and FAQ

1. Is it possible to roll non-snake-eyes forever?

There's a 0% probability of that but it's possible in the technical math sense. If, hypothetically, we had an infinite urn and rolled non-snake-eyes an inifinite number of times, we'd end up with an infinite number of inflated balls outside the urn.

2. What if the probability is undefined?

I don't think it will be but if I'm wrong, this would resolve to NO. (Unlike in the original where I accidentally created a meta-paradox and it's possible, but would still need to be debated, that the fairest resolution would be N/A.) Again, for this market, any answer other than 1/36, including "undefined", means we resolve to NO.

3. Is this different from the original snake eyes paradox?

We'll discuss in the comments. (I keep reconfusing myself on whether it is or isn't!)

4. What if the answer is "it depends"?

Then we discuss in the comments and amend this list to make that not be the answer. So you should ask any clarifying questions I've missed before trading!

Manifold Markets · Accepted Answer

No — resolved on Sep 2, 2023 by Manifold Markets prediction market.