How is it possible to win one-in-infinity odds?

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Discussion Overview

The discussion revolves around the concept of winning with one-in-infinity odds, particularly in the context of a hypothetical roulette wheel with an infinite number of outcomes. Participants explore the implications of such odds, questioning the nature of probability when dealing with infinite sets and the distinction between zero probability and impossibility.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants argue that one-in-infinity odds can be theoretically possible, while questioning if such odds are effectively zero.
  • Others propose that in a countably infinite scenario, it is feasible to assign positive probabilities to individual outcomes as long as their sum equals one.
  • There is a distinction made between countably infinite and uncountably infinite sets, with some participants noting that uncountably infinite outcomes cannot all have positive probabilities.
  • Some participants challenge the idea that real experiments can yield infinite precision, suggesting that practical outcomes are limited to finite or countably infinite possibilities.
  • Concerns are raised about the ability to communicate or measure real numbers in a practical experiment, questioning the feasibility of generating outcomes from an uncountable set.
  • Participants discuss the implications of mathematical constructs versus physical reality, particularly in relation to random variables and infinite sequences.
  • There is a debate over whether certain outcomes can be ruled out as impossible in a real-world context, especially regarding the selection of points from continuous intervals.

Areas of Agreement / Disagreement

Participants express multiple competing views on the nature of probability in infinite scenarios, and the discussion remains unresolved regarding the feasibility of experiments yielding infinite precision or outcomes from uncountably infinite sets.

Contextual Notes

Limitations include the dependence on definitions of infinity, the nature of probability distributions, and the unresolved mathematical implications of measuring outcomes from continuous sets.

  • #61
PeroK said:
@Dale can you suggest an experiment that would produce any real number (in an interval, say) and prove (or at least justify) why any real number could result.

I assume you accept that most real numbers are indescribable (uncomputable), and the set of computable numbers is countable with measure zero. From that point of view, your experiment could at best claim to have chosen a real number, but could not specify which one. And, in particular, if two such experiments were carried out there would be no algorithmic way to test whether the numbers are equal.

This is a key paradox of the real numbers. We can test mathematically that ##x = y##, where ##x, y \in \mathbb R##. But, there is no terminating algorithm to check whether two real numbers are equal. Unless you restrict things to the computable subset. IMO, that is a good example of where a simple piece of mathematics (If ##x = y \dots##), is not actually physically/algorithmically possible.
Exactly. I don't think people understand how nasty real numbers and that most are uncomputable.
 
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  • #62
Dale said:
Are you sure about that? I think you are maybe claiming that we cannot measure an element of a continuum. But I am not sure that is true.
Forget about measuring a point on a continuum - we'll never get to spin the wheel.

As soon as I ask people to place their bets, I will need enough paper (or other material) to record their selections.
If I number the bins in decimal and the players record their selections on pieces of paper, how many digits does each piece of paper need to hold?
 
  • #63
If there is a such thing as a real-valued measurement then bets can be placed the same way. E.g. if marking a piece of paper is real-valued then bets can be submitted as marked pieces of paper.

Anyway, you are now the third person that I have had to tell that I don’t find their arguments convincing. Again, I also have not thought about this carefully myself and don’t have contrary arguments that I find convincing yet. Until I have convinced myself, all I have is doubts which I am done arguing. Please don’t try to draw me in again. I will open a new thread when I am ready.
 
  • #65
After moderator review, the thread will remain closed.
 

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