Do Infinity and Statistics Always Go Hand in Hand?

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Discussion Overview

The discussion revolves around the relationship between infinity and statistics, specifically focusing on the implications of rolling a die an infinite number of times and the concept of "almost sure" outcomes in probability. Participants explore theoretical scenarios involving infinite trials and their statistical outcomes, including the classic example of infinite monkeys typing Shakespeare.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that rolling a die an infinite number of times would eventually result in rolling a 6, suggesting that the probability of this outcome is 100%.
  • Others argue that while the probability of rolling a 6 is 1, this does not guarantee that it must happen, as it is possible to roll a die infinitely and never roll a 6.
  • A participant mentions that in mathematics, the phrase "almost surely" is used to describe the situation where an outcome has a probability of 1, but does not imply certainty in every instance.
  • There is a discussion about the implications of continuous probability distributions and how they relate to infinite trials, highlighting that a probability of 0 does not mean an event cannot occur.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of probability in the context of infinite trials. While some agree on the concept of "almost sure" outcomes, others emphasize the nuances that prevent definitive conclusions about specific occurrences.

Contextual Notes

Participants acknowledge the idealized nature of the scenarios discussed, noting that real-life conditions (such as the fairness of dice) may not align with theoretical assumptions.

red apple
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Hi there, I have a question regarding infinity and statistics. (I hope there aren't too many questions with infinity on these forums)

I was wondering if you had some simple procedure, like say rolling a six sided die, and said you did this an infinite amount of times, would it be valid to say eventually you will roll a 6? Of course this is an ideal situation where each side has an equal chance of being rolled.

I'm wondering if it is accurate to say you will eventually have any outcome occur (assuming it is possible on any individual trial, no matter how unlikely)

Is this even valid considering its not really possible to actually have this happen in real life?

Also just for fun, what about an example taken to the extreme like an infinite amount of monkies all writing on typewriters, assuming they at least type something will at least one type a Shakespeare play exactly? Will an infinite amount type a Shakespeare play?

Thank you.
 
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red apple said:
I was wondering if you had some simple procedure, like say rolling a six sided die, and said you did this an infinite amount of times, would it be valid to say eventually you will roll a 6? Of course this is an ideal situation where each side has an equal chance of being rolled.

Not really, but you can say that the chance of rolling a 6 is 100% (exactly).

red apple said:
Is this even valid considering its not really possible to actually have this happen in real life?

Sure. We don't have perfectly fair dice in real life, either.

red apple said:
Also just for fun, what about an example taken to the extreme like an infinite amount of monkies all writing on typewriters, assuming they at least type something will at least one type a Shakespeare play exactly? Will an infinite amount type a Shakespeare play?

With probability 1, yes.
 
Note, however, that when you are dealing with something happening an infinite number of times (or working with continuous probability distributions), a probability of 1 does NOT mean it must happen nor does a probability of 0 mean it won't happen. It is possible to roll a die an infinite number of times and NOT get a "6". It is even possible to roll a die an infinite number of times and always get a "2" (Assuming that it is possible to roll a die and infinite number of times!) even though the probability of rolling a "6" is 1 and the probability of rolling all "2"s is 0.
 
Quite correct, Halls. I should have pointed that out.
 
CRGreathouse said:
Not really, but you can say that the chance of rolling a 6 is 100% (exactly).

I believe in mathematics we say that you will almost surely roll a 6.

http://en.wikipedia.org/wiki/Almost_surely
 
Diffy said:
I believe in mathematics we say that you will almost surely roll a 6.

http://en.wikipedia.org/wiki/Almost_surely

I say both "...with probability 1." and "...almost surely.", though I prefer the former. Only pedagogically do I say "The chance of ... is 100%."
 

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