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02/15/2011 06:31:47 PM · #251 |
Originally posted by DrAchoo:
But wait, how can that be, because there's also the possibility that they type the letter "t" every single time. What does that say about your all 'S' scenario? |
It proves the point! Not every outcome is possible because some totally preclude others. |
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02/15/2011 06:33:04 PM · #252 |
Originally posted by scalvert:
Originally posted by DrAchoo:
But wait, how can that be, because there's also the possibility that they type the letter "t" every single time. What does that say about your all 'S' scenario? |
It proves the point! Not every outcome is possible because some totally preclude others. |
Aha. I think the issue is whether we are looking at the odds of scenario or the odds of a scenario of scenarios. I'll keep pondering while seeing patients.
OR this: You talk about the instance where the monkeys type "S" an infinite amount of times, but the reality is those very monkeys could still type Shakespeare. Why? Because we have an infinite more number of tries by those very monkeys. Well, you say, they type "S" those infinite number of times too. OK, well, they STILL have an infinite number of attempts left. The only way you can get out of this is to fatalistically say they will ALWAYS type that S. And if you do that, you change the scenario.
Yes, that is the answer.
So, in essense, you are asking "What are the odds of typing Shakespeare if monkeys type 's' an infinite number of times?" The answer is zero.
Message edited by author 2011-02-15 18:39:41. |
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02/15/2011 06:47:24 PM · #253 |
'Always typing S' is as valid an outcome as 'Typing Hamlet', we're not changing the scenario.
The monkey sits at a typewriter, and starts randomly hitting keys. That's our scenario.
Each keystroke event is independent of the preceding one. Each time the monkey hits a key, he has a 1-in-26 (or whatever) chance of hitting the letter 's'
So whether it's his first keystroke, or his millionth, it has an equal chance of being 's' - It doesn't matter how long he sits there hitting keys. You could observe him forever, and he might never hit any other letter.
We are giving him an infinite time to type Hamlet, it's only fair we give him an infinite time to hit the 's' key. They both have an equal chance of happening. But the main point of what Mathis is saying is that while the possibility of hitting 's' for infinity exists (and it does exist) then we can't give 100% certainty to any other outcomes.
Message edited by author 2011-02-15 18:48:36. |
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02/15/2011 06:51:42 PM · #254 |
Originally posted by DrAchoo:
I think the issue is whether we are looking at the odds of scenario or the odds of a scenario of scenarios. |
Odds of a scenario = probability. You're making the faulty assumption that because the probability of a given outcome isn't absolutely zero, every scenario must occur given infinite opportunity. However, since the probability of a given outcome NOT occurring is also not zero (and some outcomes preclude others), the assumption cannot be true.
If you flip a quarter a bunch of times, what is the probability that one of those times a house sparrow will catch it in midair and then get whacked by a meteorite, dropping the quarter on edge upon the back of an albino box turtle? Extremely low, but not zero. If you flip the quarter an infinite number of times, is this scenario guaranteed to happen even once? Nope. |
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02/15/2011 06:53:22 PM · #255 |
| I saw that happen once on acid. |
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02/15/2011 07:05:23 PM · #256 |
Originally posted by JH:
'Always typing S' is as valid an outcome as 'Typing Hamlet', we're not changing the scenario.
The monkey sits at a typewriter, and starts randomly hitting keys. That's our scenario.
Each keystroke event is independent of the preceding one. Each time the monkey hits a key, he has a 1-in-26 (or whatever) chance of hitting the letter 's'
So whether it's his first keystroke, or his millionth, it has an equal chance of being 's' - It doesn't matter how long he sits there hitting keys. You could observe him forever, and he might never hit any other letter.
We are giving him an infinite time to type Hamlet, it's only fair we give him an infinite time to hit the 's' key. They both have an equal chance of happening. But the main point of what Mathis is saying is that while the possibility of hitting 's' for infinity exists (and it does exist) then we can't give 100% certainty to any other outcomes. |
But this is the magic and maddening nature of infinity. You could still always have the possibility of typing Hamlet. You see? We say he types S for infinity. Well, what does he type next? The odds are 1-26 right that he could type the first letter of the first word. In other words, you can never declare that he types 's' an infinite number of times because he would never get to the end of the infinite progression. When would you be able to declare, a-ha!, this is that time that foils those Hamlet jerks? You never would. Because he'd still have an infinite number of random attempts left and in those attempts he could type Hamlet.
This is why I get queasy thinking about infinity.
Message edited by author 2011-02-15 19:06:56. |
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02/15/2011 08:03:12 PM · #257 |
Originally posted by DrAchoo:
When would you be able to declare, a-ha!, this is that time that foils those Hamlet jerks? You never would. Because he'd still have an infinite number of random attempts left and in those attempts he could type Hamlet. |
There you have it - 'could', not 'will'
The point is that just as he could continue typing 's' for infinity, he also could sit down and type Hamlet on his first attempt. We can't say that he definitely will without doubt and with 100% certainty type Hamlet if he's given an infinite length of time. |
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02/15/2011 09:54:29 PM · #258 |
Originally posted by Louis:
I saw that happen once on acid. |
You too? Small world... |
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02/16/2011 12:45:52 AM · #259 |
Originally posted by JH:
I can see it now.
Monkey number 10^8273 sitting at his typewriter. He's looking tired as he's just spent the past two weeks randomly hitting keys, but unbeknown to him has actually produced all the words of Hamlet. All of them, except one. In fact, he's on the last letter of the last word; "off". His finger hovers over the "f"..
A universe-worth of cosmologists, philosophers, and mathematicians hold their collective breaths. |
There's an old sci-fi story about a group of scientists who actually set up an experiment using a bunch of monkeys with typewriters, with somewhat disturbing results ... |
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02/16/2011 02:20:47 AM · #260 |
| I'm just going to have to go with my instinct and go with Greene. Call me crazy... |
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02/16/2011 02:54:50 AM · #261 |
Originally posted by DrAchoo:
I'm just going to have to go with my instinct and go with Greene. Call me crazy... |
A monkey typing out Hamlet is probably more likely to occur than you admitting defeat.
Message edited by author 2011-02-16 02:55:06.
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02/16/2011 07:24:28 AM · #262 |
Originally posted by DrAchoo:
So you just have to make up your mind when you're talking to me. Are the chances zero? or are they just really, really small? The answer is completely different depending on which you are talking about. Monkeys typing out shakespeare falls into the very small category, not the zero category. If you are limiting them by the age of the universe you are just not giving them infinite iterations... |
I was talking at cross purposes with you as to the meaning of "monkeys" in the analogy. If by "monkeys" we mean only random letter generators, then it is not impossible for random letter generators to come up with Shakespeare (though it is amazingly unlikely).
As I said before, I should have made it more clear in my throwaway comment that I perceived monkeys more literally as a chaotic system of random letter generation in which noise would always persist.
Complex systems are chaotic - and the universe is just about the most complex system imaginable. It is one thing to calculate the odds of a finite string appearing within a limited variable system, and quite another to conceive of an infinitely complex variable within a universe of infinite complexity.
Once again: in a complex system, given infinite time you will not get every conceivable situation arising as a matter of chance. Impossible things will not happen just because there are many or infinite iterations. Edit to add (after reading some of the other posts): and it remains against the odds that very unlikely things would necessarily ever happen.
Message edited by author 2011-02-16 07:35:19.
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02/16/2011 08:08:45 AM · #263 |
Infinity is actually a whole study for itself. A problem in studying it is that it does not exist in any tangible form that enables empirical observation. It is, in fact, an idea or concept of which the only practical application is in creating and/or enabling mathematical models.
Nought point 9 recurring is equal to 1. It is not 'nearly equal' or 'as near as makes no odds'. It is precisely the same as one.
Given infinite time, everything happens.
This makes the doc. queezy, so it might be better not to talk about it.
It is at least mildly fatuous to talk of the probability of an event in a time frame without parameters - or without a time frame, to put it another way. All bets are off.
The chances of any being or thing evolving into something and knocking out the works of shakespeare along with a good few other pearls the bard never quite got round to, in an infinite time span, are 100% |
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02/16/2011 08:38:03 AM · #264 |
Originally posted by raish:
The chances of any being or thing evolving into something and knocking out the works of shakespeare along with a good few other pearls the bard never quite got round to, in an infinite time span, are 100% |
It's actually zero percent. The model rests on several dubious assumptions that must all be true.
1: Infinite time spans are possible (time apparently began at a finite point)
2. The outcome is possible (a cow will never jump over the moon)
3. Monkeys are random generators. (they're not) |
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02/16/2011 09:13:53 AM · #265 |
Originally posted by scalvert:
If you flip a quarter a bunch of times, what is the probability that one of those times a house sparrow will catch it in midair and then get whacked by a meteorite, dropping the quarter on edge upon the back of an albino box turtle? Extremely low, but not zero. If you flip the quarter an infinite number of times, is this scenario guaranteed to happen even once? Nope. |
Happened in New England yesterday. Damn that was a bright meteor. I only flipped the quarter once. A friend wanted
to make soup out of the turtle. We flipped for it. He got the house sparrow. |
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02/16/2011 10:58:46 AM · #266 |
Originally posted by raish:
The chances of any being or thing evolving into something and knocking out the works of shakespeare along with a good few other pearls the bard never quite got round to, in an infinite time span, are 100% |
You are correct raish. I was thinking about it while playing poker last night and I am convinced it is correct. (unless people start changing the scenario). The chaotic system idea is bunk though.
Think about it this way. One monkey who types letters randomly with an infinite time limit. You start the experiment and watch. If he types Hamlet, you stop and the experiment is a success. If you check and he hasn't done it, he keeps going.
What is the probability the experiment will end in failure? The answer is zero. The experiment only ends if it is successful, otherwise it keeps going. That's the magic of infinity. So, conversely, what is the probability the experiment will end in success? Since we subtract zero from 1, we get our answer of 1.
Message edited by author 2011-02-16 10:59:35. |
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02/16/2011 11:02:17 AM · #267 |
Originally posted by DrAchoo:
Originally posted by raish:
The chances of any being or thing evolving into something and knocking out the works of shakespeare along with a good few other pearls the bard never quite got round to, in an infinite time span, are 100% |
You are correct raish. I was thinking about it while playing poker last night and I am convinced it is correct. (unless people start changing the scenario). The chaotic system idea is bunk though.
Think about it this way. One monkey who types letters randomly with an infinite time limit. You start the experiment and watch. If he types Hamlet, you stop and the experiment is a success. If you check and he hasn't done it, he keeps going.
What is the probability the experiment will end in failure? The answer is zero. The experiment only ends if it is successful, otherwise it keeps going. That's the magic of infinity. So, conversely, what is the probability the experiment will end in success? Since we subtract zero from 1, we get our answer of 1. |
You neglect the possibility that the experiment never ends. Part of the magic of inifinity is that simple arithmetic (your 1-0) doesn't work with it. The experiment itself can become infinite with no resolution. |
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02/16/2011 11:27:08 AM · #268 |
Originally posted by eqsite:
Originally posted by DrAchoo:
Originally posted by raish:
The chances of any being or thing evolving into something and knocking out the works of shakespeare along with a good few other pearls the bard never quite got round to, in an infinite time span, are 100% |
You are correct raish. I was thinking about it while playing poker last night and I am convinced it is correct. (unless people start changing the scenario). The chaotic system idea is bunk though.
Think about it this way. One monkey who types letters randomly with an infinite time limit. You start the experiment and watch. If he types Hamlet, you stop and the experiment is a success. If you check and he hasn't done it, he keeps going.
What is the probability the experiment will end in failure? The answer is zero. The experiment only ends if it is successful, otherwise it keeps going. That's the magic of infinity. So, conversely, what is the probability the experiment will end in success? Since we subtract zero from 1, we get our answer of 1. |
You neglect the possibility that the experiment never ends. Part of the magic of inifinity is that simple arithmetic (your 1-0) doesn't work with it. The experiment itself can become infinite with no resolution. |
Ok, let's calculate that probability.
One monkey. Types letters randomly. If he types Hamlet, we stop and count the experiment a failure. If he doesn't, the experiment keeps going.
What is the probability the experiment ends in failure? The answer is 1. The experiment only ends if it is a failure, otherwise it keeps going.
The trick with this all is the semantics of infinity. You say "the experiment itself can become infinite with no resolution", but when could you ever declare this? When could you hold in your hands the result of such an iteration? You can't. Because it's still going. It's a logical impossibility. Therefore the experiment always ends with Hamlet. |
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02/16/2011 11:42:32 AM · #269 |
Originally posted by DrAchoo:
Originally posted by eqsite:
You neglect the possibility that the experiment never ends. Part of the magic of inifinity is that simple arithmetic (your 1-0) doesn't work with it. The experiment itself can become infinite with no resolution. |
Ok, let's calculate that probability.
One monkey. Types letters randomly. If he types Hamlet, we stop and count the experiment a failure. If he doesn't, the experiment keeps going.
What is the probability the experiment ends in failure? The answer is 1. The experiment only ends if it is a failure, otherwise it keeps going.
The trick with this all is the semantics of infinity. You say "the experiment itself can become infinite with no resolution", but when could you ever declare this? When could you hold in your hands the result of such an iteration? You can't. Because it's still going. It's a logical impossibility. Therefore the experiment always ends with Hamlet. |
No, the experiment may never end. The monkey may keep typing and typing without ever reaching the desired outcome. You assume that given enough iterations, every possible outcome will eventually come out. This is not true. Every possible outcome has a probability of occuring but this does not mean that it will eventually occur. If, as you seem to be asserting, given enough iterations, every possible outcome will eventually occur, then the probably of a given event will change (approach 1) as the number of iterations increases. This is not the case. The probability remains the same regardless of the number of iterations. Running the experiment infinitely does not guarantee the result, as the probability does not change. |
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02/16/2011 12:15:09 PM · #270 |
| People win the Lotto 6/49 by randomly matching six numbers between zero and 49. The odds are very high, but it happens all the time... |
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02/16/2011 12:21:58 PM · #271 |
Originally posted by Strikeslip:
People win the Lotto 6/49 by randomly matching six numbers between zero and 49. The odds are very high, but it happens all the time... |
Yes, but playing the same numbers over and over again are no guarantee of winning. |
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02/16/2011 12:28:40 PM · #272 |
Originally posted by eqsite:
No, the experiment may never end. The monkey may keep typing and typing without ever reaching the desired outcome. You assume that given enough iterations, every possible outcome will eventually come out. This is not true. Every possible outcome has a probability of occuring but this does not mean that it will eventually occur. If, as you seem to be asserting, given enough iterations, every possible outcome will eventually occur, then the probably of a given event will change (approach 1) as the number of iterations increases. This is not the case. The probability remains the same regardless of the number of iterations. Running the experiment infinitely does not guarantee the result, as the probability does not change. |
What he said. A cow trying to jump over the moon is not guaranteed success no matter how many attempts are made, and it doesn't matter if the experiment continues forever. Same goes for my earlier example- it's possible for a sparrow to catch catch a quarter, it's possible for a meteorite to hit a sparrow, it's possible for a quarter to drop and land on its edge, and it's possible for it to land on an albino box turtle. However, that will NEVER occur in sequence no matter how many opportunities are given. Moreover, the apparently finite beginning of time and ultimate heat death of the universe mean the experiment cannot continue forever.
For most practical purposes, stars are considered fixed points at an infinite distance. Limitations on the speed of information and the expansion of the universe may mean you could never reach a remote star no matter how long you travel, yet the distance is not really infinite. It's only a mathematical convenience. |
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02/16/2011 12:40:52 PM · #273 |
Sorry guys. I am correct. I don't know how else to make you see it.
The proof is in the seeing. Right? When are you ever going to "see" an infinite series that never types Hamlet. When can you declare, "I've seen it with my own eyes!"? Never. You never can because the series never ends. You would have to be able to "see" to the end of infinity and that is a logical impossibility. |
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02/16/2011 12:45:58 PM · #274 |
Originally posted by GeneralE:
Originally posted by JH:
A universe-worth of cosmologists, philosophers, and mathematicians hold their collective breaths. |
There's an old sci-fi story about a group of scientists who actually set up an experiment using a bunch of monkeys with typewriters, with somewhat disturbing results ... |
I was trying to dig that one up... Haven't found it yet...
R. |
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02/16/2011 12:48:59 PM · #275 |
Originally posted by DrAchoo:
Sorry guys. I am correct. I don't know how else to make you see it. |
Wow. Just wow. |
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