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Author
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Topic: The Uses of Limits in Probability
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Regvi
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Member # 586
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posted 06. December 2002 20:28
Some statisticians claim that the probability of getting infinitely many heads after tossing infinitely many coins is nil. The argument is this: in mathematical analysis, x/infinity = 0.
But I claim that this is a wrong application of the principle of classical probability, which states that the probability of an event is "the number of advantageous outcomes/the number of possible outcomes", eg the probability of gettting 4 heads in a row is 1/32.
The problem is that 1) getting all heads is a possible outcome, and therefore its probability should not be 0, and 2) that the sum of the probabilities doesn't sum to 1, but to 0.
It's also easy to see that it's impossible to assign any probability to an infinite sequence of coin tosses, since the sum will become infinite, not 1.
[ 07. December 2002, 21:34: Message edited by: Moderator ]
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Evan
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posted 06. December 2002 21:58
This topic is probably not going to fly as an opening post here at ISCID, but I think you don't understand the idea of limits as applied to infinity.
The correct way of stating what you are talking about is that "1/n goes to zero as n goes to infinity," so we say "the limit of 1/n is zero as n goes to iinfinity." Infinity is not a place you ever get, and therefore the probability never gets to zero. But we can get it as close to zero as we like if we just let n get large enough - that is the essence of epsilon - delta limits proofs in calculus.
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Moderator
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posted 06. December 2002 23:39
I've re-opened this topic. However, as I said before ...
Telling us why someone else is wrong just isn't going to cut it. Threads should not be started as critiques if there is no positive hypotheses put forward.
I've requested the following from the thread starter:
1. Draw out the implications for a non mathematical audience: implicitly
tell others why this thread is important
2. If you are going to say that someone is wrong, then spend proportionally much more time arguing for and defending your own position
[ 07. December 2002, 21:36: Message edited by: Moderator ]
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Regvi
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posted 08. December 2002 13:53
This topic doesn't have much practical significance. But it's a good way to learn more about probability and mathematical analysis.
Consider tossing a coin 5 times. The probability of getting only heads is then
p(n heads) = 1/(2.2.2.2.2)
As n gets bigger and bigger, p(n heads) gets arbitrarily close to 0. But it never becomes exactly 0.
Consider tossing a coin infinitely many times. What is the probability of getting only heads now? It turns out that it's impossible to assign a probability to this event, for the reasons I stated in my first posting. Informally we may say that p(only heads) is 0, since there are infinitely many possible outcomes and only one advantageous. In this case there is a good fit between the limit as n goes to infinity and the possibility of getting infinitely many heads.
Now consider the "random walk" experiment. Starting at 0, we go one step left or one step right on the x-axis if we get heads or tails. As the number of coin flips n increases, the probability of returning to 0 converges to 1. What happens if we flip a coin infinitely many times? How big are our chances of returning to 0? As I have already mentioned, it's impossible to calculate a probability. But the above limit seems to suggest that the chances are very good. But this is wrong: there are infinitely many sequences which return to 0. But there are also infinitely many sequences which don't return to 0. Therefore, it's difficult to say how good the chances are for infinitely many coin flips, even if the chances become better and better for larger and larger n. This may be somewhat surprising.
On the basis of what I have said above, I want to suggest that finite and infinite probabilities be treated separately. Finite probabilites may be calculated; infinite ones may not. If the number of advantageous outcomes in the infinite case is finite, then it makes good sense to speak of the 'probability' of it as 0. But if the number of advantageous outcomes is infinite, then there doesn't seem to be any way of assessing the 'probability' of the event. But perhaps someone else knows a method which can be used.
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Evan
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posted 08. December 2002 21:33
Regvi writes, “this topic doesn't have much practical significance. But it's a good way to learn more about probability and mathematical analysis.”
Given that one of my interests in design theory has to do with assigning probabilities to events, this topic is somewhat pertinent. I think Regvi is still confused about a few things, and I’d like to clear them up.
First let me say that Regvi does correctly discuss the reasons why we say that “informally we may say that p(only heads) is 0” for an infinite number of flips But we always must remember that you can’t really “flip a coin an infinite number of times” - all we can do is explore the behavior of n flips as n gets bigger and bigger.
Secondly, in respect to the “random walk” Regvi says,
quote:
Starting at 0, we go one step left or one step right on the x-axis if we get heads or tails. As the number of coin flips n increases, the probability of returning to 0 converges to 1.
This is wrong. As the number of flips increases, the probability of returning to 0 (I assume meaning at the end of the sequence) approaches 0, not 1. If you flip a coin 1,000,000 times, the probability of having exactly 500,000 heads is extremely small, and it is vastly smaller still if you flip a coin 1 billion times and expect 1/2 billion heads.
Then, Regvi writes, “On the basis of what I have said above, I want to suggest that finite and infinite probabilities be treated separately. Finite probabilities may be calculated; infinite ones may not. If the number of advantageous outcomes in the infinite case is finite, then it makes good sense to speak of the 'probability' of it as 0. But if the number of advantageous outcomes is infinite, then there doesn't seem to be any way of assessing the 'probability' of the event. But perhaps someone else knows a method which can be used.”
All of these topics are covered in statistics. For all real-world purposes, there are a finite number of probabilities, and events are treated as discrete. But for many purposes it is much easier to consider a continuous probability function and use calculus to do the work. In that case, there are times where one might model the situation so that there was an infinite number of events and an infinite number of advantageous outcomes, and still find a probability using standard methods of finding a limit.
A simple example is this: pick a number (an integer, not zero). The odds are 50% that you picked an even number, despite the fact that you picked from an infinite set and you had an infinite set to pick from.
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Regvi
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posted 09. December 2002 06:54
**First let me say that Regvi does correctly discuss the reasons why we say that “informally we may say that p(only heads) is 0” for an infinite number of flips But we always must remember that you can’t really “flip a coin an infinite number of times” - all we can do is explore the behavior of n flips as n gets bigger and bigger.
Reply: I disagree. At the very least it's possible to imagine tossing an infinite number of coins simultaneously and thus getting an infinite sequence.
**Secondly, in respect to the “random walk” Regvi says,
quote:
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Starting at 0, we go one step left or one step right on the x-axis if we get heads or tails. As the number of coin flips n increases, the probability of returning to 0 converges to 1.
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This is wrong. As the number of flips increases, the probability of returning to 0 (I assume meaning at the end of the sequence) approaches 0, not 1. If you flip a coin 1,000,000 times, the probability of having exactly 500,000 heads is extremely small, and it is vastly smaller still if you flip a coin 1 billion times and expect 1/2 billion heads.
Reply: You misunderstood me. I meant the probability of returning at any point, not necessarily at the end of the sequence. This probability converges to 1.
**Then, Regvi writes, “On the basis of what I have said above, I want to suggest that finite and infinite probabilities be treated separately. Finite probabilities may be calculated; infinite ones may not. If the number of advantageous outcomes in the infinite case is finite, then it makes good sense to speak of the 'probability' of it as 0. But if the number of advantageous outcomes is infinite, then there doesn't seem to be any way of assessing the 'probability' of the event. But perhaps someone else knows a method which can be used.”
All of these topics are covered in statistics. For all real-world purposes, there are a finite number of probabilities, and events are treated as discrete. But for many purposes it is much easier to consider a continuous probability function and use calculus to do the work. In that case, there are times where one might model the situation so that there was an infinite number of events and an infinite number of advantageous outcomes, and still find a probability using standard methods of finding a limit.
A simple example is this: pick a number (an integer, not zero). The odds are 50% that you picked an even number, despite the fact that you picked from an infinite set and you had an infinite set to pick from.
Reply: As far as I know, it's theoretically impossible to assign equal probabilities to the set of natural numbers. But I have another example: consider an infinite sequence of coin tosses (I have already argued that it's logically possible to find one). Then the chances of getting one that starts with "Heads" should be equal to the probability of getting one that starts with "Tails".
Even so, there may be cases where it's impossible to assign probabilities, as in the random walk experiment with infinitely many coin tosses.
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Evan
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posted 10. December 2002 11:19
There are some interesting issues here.
First, Regvi writes, “consider an infinite sequence of coin tosses (I have already argued that it's logically possible to find one). Then the chances of getting one that starts with "Heads" should be equal to the probability of getting one that starts with "Tails".[/quote]
This is true for any set of coin tosses. This fact has nothing to do with how many coin tosses follow.
Also, Regvi also writes, “I meant the probability of returning at any point, not necessarily at the end of the sequence. This probability converges to 1.”
I still don’t think this is true. Once a random walk diverges from the origin, the chances of it returning to the origin are definitely not 1.
There’s a nice 2-d simulation of this at math.furman.edu/~dcs/java/rw.html
Select something like 20,000 trials. Push draw, and then push continue after each subsequent draw. You will get a graphic picture of what I’m talking about. You might return to the origin, but I’m virtually certain that there will be more times that you don’t. I remember that there are some proofs of this, but I’ve been unable to locate them in a first quick glance through my various calculus books.
I am not trying to be argumentative here, and will be happy to be shown to be wrong.
But i know it is easy to have wrong ideas about probabilities and randomness, and since those topics are central to design arguments, I think it’s important to pursue correct understandings.
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Regvi
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posted 10. December 2002 14:06
The random walk I was describing took place in 1-d space, ie on the x-axis. It seems that the probability of ever returning to the origin converges to 0, not 1 as I claimed.
Here's the recursive function which says how many possible ways there are to return to the origin after exactly n steps, where (2 1) means "2 choose 1":
w(2) = (2 1)
w(4) = (4 2) - w(2).(2 1)
w(6) = (6 3) - w(2).(4 2) - w(4).(2 1)
w(8) = (8 4) - w(2).(6 3) - w(4).(4 2) - w(6).(2 1)
Empirically, it seems that the probability of ever returning to the origin converges to 0.
But what happens in 2 dimensions? According to "Data Structures & Problem Solving Using Java" by M.A. Weiss, in this case the probability converges to 1. But then my main point can be made: if we consider infinitely many steps, then there are infinitely many sequences which never return to the origin. Therefore, I think it's impossible to calculate any probability when the sequence is infinite, in contrast to the finite cases, where the probability converges to 1. Therefore, as I have already said, there is a jump from the finite to the infinite cases.
You mention simulating the random walk. This can be a great help in testing whether or not our calculations are correct. I'll have a look at it and maybe write some comments on it.
[ 10. December 2002, 15:32: Message edited by: Regvi ]
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