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Comment by RiderOfGiraffes | Hacker News Reader

People really, really don't understand the concept of infinity, nor the concept of "with probability 1." The most interesting point about the idea that people actually go out and try these things, is to prove that people just don't understand these things.

The conclusion that "the theory is flawed" shows they don't understand the theory.

The question I have is - should we care? Should we try to explain these things? Or should we just shrug and give up. How will it actually make their lives better? For some people it might, but then again ...

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If the probability of producing the desired result in any given attempt is zero, then the probability of producing the desired result given infinite attempts is still zero.

If monkeys are incapable of typing, then the probability is zero.

  > If the probability of producing the desired result
  > in any given attempt is zero, then the probability
  > of producing the desired result given infinite
  > attempts is still zero.

This is a common misconception. Pick a point uniformly at random in the interval from 0 to 1. The probability of any given point is zero, but the probability of being in the interval [a,b) is b-a (assuming 0<=a<=b<=1). This is a case of infinitely many zeroes giving a non-zero sum, and why you need to deal carefully with infinities.

Yes, I know about countable and uncountable infinities. This is a big subject, and small comments here won't cover it. There are rules and exceptions.

One over infinity is a different kind of zero than we're talking about. An infinitesimal value is not a nil value.

One over infinity is not a zero at all, because it's an ill-defined operation. You need to say exactly what you mean, and which number system you're working in. Are you in the surreals? Non-standard analysis? The hyper-reals?

If you work with standard analysis, your comment seems to be a non-sequiteur. If not, you need to say which system you're in. Personally, I think the other systems are largely unhelpful, although independently interesting.

And of course an infinitesimal is not zero (nil being a different technical term).

If I've missed the point(s) you were tying to make, you'll need to be more explicit. Probably further "discussion" here is not going to be fruitful, but I'll be interested to read anything you care to add.

I'm sorry, I'm not a statistician (or any kind of mathematician) so I don't know the proper terminology. I'm just noting that the only way your previous example seems work from my perspective is if you take one over infinity to be zero.

I'll explain: The set of all of the points between two points is an infinite set, and the probability of selecting any single member of a set is approximately one over the size of the set, so it would seem that the probability of selecting any individual member of an infinite set would be one over 'infinity' that is, an undefined value.

Granted, the limit of one over x as x approaches infinity is zero, but the probability of selecting any single member of an infinite set isn't zero in the same sense that the probability of selecting a member outside the set is zero. That is, if I pick a point between 0 and 1, and I select point x, there was obviously not impossible to select point x. So while it may be 'mathematically convenient'* to say that the probability of selecting any given point is zero I think that it might be better to say that it might be better to say that the chance is infinitesimally small, rather than zero per se.

So while I may certainly be wrong, I think that you are univocating when you say that a range is an example of an infinite number of zeros being non-zero. When I was talking about the chance of a individual monkey typing out Shakespeare was zero, I meant that, if it was impossible for any monkey to type (imagine that we are using exclusively dead monkeys) no monkey could ever type out anything, much less Shakespeare.

The problem is that if you assign a probably to a point, then the probability you assign has to be strictly smaller that every positive number, so it has to be zero. The system that works with finite sets of taking the number of things you're interested in and dividing by the number of things that might happen doesn't apply. Contradictions and/or inconsistencies arise from that sort of thinking, seductive though it is.

My example about the ranges was an attempt to show that the otherwise excellent ideas from the finite world simply don't apply, so you have to do something else. You suggested that ...

  > If the probability of producing the desired result
  > in any given attempt is zero, then the probability
  > of producing the desired result given infinite
  > attempts is still zero.

... and that's what I was refuting.

And perhaps the things to take away from this is that just because the probability of an event is zero, that doesn't mean the event can't happen. Talking about the probability being "infinitesimally small" requires that you make a whole pile of other definitions precise, and that way lies madness. Or at least, non-standard analysis, which is pretty close to the same thing.

In my opinion it doesn't help the understanding of the "interested layman."

All this, of course, is only valid in the theoretical musings of mathematicians (or those doing math at the time) In the "real world" it's all different again.

Doing this sort of thing correctly and consistently is hard.

And perhaps the things to take away from this is that just because the probability of an event is zero, that doesn't mean the event can't happen.

Ah! That does seem to be the take-away here. In that case, however, how would you measure the probability of an impossible event?

I still maintain that an invinite number of dead monkeys would never type Shakespeare, no matter the resources alloted to them. :)

Yes this small study does not disprove that, given enough monkeys and enough typewriters, a shakespearean play will be produced.

However, we need only look to the internet to see that the proposition is indeed false.

Cute, but wrong (as you probably know).

Even the size of the internet and all the drivel on it is infintesimally smaller than infinity.

Ron Graham once told me that the problems we have in mathematics are caused in part by the fact that we only ever deal with small numbers, and he's the one who used Graham's Number.

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

On a serious note, have people been more educated in probability, the economic crisis might not have turned out as bad. Look at Nassim Taleb to see how far some probability consciousness can get you.

While I certainly agree with you, I must admit that it never occurred to me that the whole monkey typist experiment depends on the assumption of the monkey's being able to actually hit each key. :)

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