Statistics with infinity

[red apple]

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.

 

[CRGreathouse]

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).

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.

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.

 

[HallsofIvy]

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.

 

[CRGreathouse]

Quite correct, Halls. I should have pointed that out.

 

[Diffy]

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

 

[CRGreathouse]

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%."