RE: The infinite list of random numbers
Yes, but think how many Tom Clancy books it would write in the mean-time. Also, think of all the mystery books with the last page re-arranged to be the first, or all those many ones with typos. -Original Message- From: Norman Samish [mailto:[EMAIL PROTECTED]] Sent: 11 November 2001 05:32 To: [EMAIL PROTECTED] Subject: The infinite list of random numbers Thanks to all who replied. Thanks to your instruction, it now is clear to me that, in an infinite series of random characters, every conceivable sequence MUST occur. These sequences must, of course, obey the requirement that all random characters in an infinite sequence must appear an equal number of times. This requirement rules out sequences of only one character. Therefore, in infinite time, the long-lived monkey at the durable typewriter HAS to eventually write the works of Shakespeare, as well as anything else conceivable. More generally, everything that can happen MUST happen, not only once but an infinite number of times. Norm Samish
Re: The infinite list of random numbers
--- scerir [EMAIL PROTECTED] wrote: rwas : To be able to say that a process will be random at infinite time would seem to imply a deterministic process that can generate non determinism. :) That deterministic process might be the construction of this true random number generator: http://www.gapoptic.unige.ch/Prototypes/QRNG/default.asp -s. Intuitively, it would seem to me that the entropy of any temporally aparently random sequence would drop over time. That is a sequence may have high entropy over some sampling interval, but then the entropy would drop over a progressively larger sample. Robert W. __ Do You Yahoo!? Find a job, post your resume. http://careers.yahoo.com
The infinite list of random numbers
Thanks to all who replied. Thanks to your instruction, it now is clear to me that, in an infinite series of random characters, every conceivable sequence MUST occur. These sequences must, of course, obey the requirement that all random characters in an infinite sequence must appear an equal number of times. This requirement rules out sequences of only one character. Therefore, in infinite time, the long-lived monkey at the durable typewriter HAS to eventually write the works of Shakespeare, as well as anything else conceivable. More generally, everything that can happen MUST happen, not only once but an infinite number of times. Norm Samish
Re: The infinite list of random numbers
Norman Samish wrote: Suppose an ideal random number generator produces, every microsecond, either a zero or a one and records it on a tape. After a long time interval one would expect the tape to contain a random mix of zeroes and ones with the number of zeroes equal to the number of ones. Is this necessarily true? Is it possible that, even after an infinite time had passed, that the tape could contain all zeroes or all ones? Or MUST the tape contain an equal number of zeroes and ones? Why? If you have a reference dealing with this topic, please let me know. Thanks, Norm Samish I don't think we can view time in terms of time passed and infinite. I think we can look at the problem in terms of a set of numbers over all time, or we can look at a set of numbers issued as a stream sampled over finite time. I think a set of numbers can only be defined as infinite over all time, not tested as such. To be able to say that a process will be random at infinite time would seem to imply a deterministic process that can generate non determinism. :) As for the ideal random number generator, if it's truly ideal, you could easily never see it produce anything buts all ones or all zeros for your lifetime, then promptly after you're dead, it starts producing something that appears random to a temporally constrained observer. An ideal random number generator could only be proved to be ideal over all time since any finite sampling of the stream would necessarily introduce order into the evaluation, lowering entropy and reducing randomness. IMO Robert W. _ Do You Yahoo!? Get your free @yahoo.com address at http://mail.yahoo.com
Re: The infinite list of random numbers
All arrangemets are equally likely, but the probability is, of course, zero. So with probability one you don't get only zeros. There is a theorem that says that any finite arbitrary configuration will appear an infinite number of times in an infinite random sequence with probability one. Saibal Neil Lion wrote: It's undefinable. You're just as likely to get all zeros, or all ones, as you are to get any arrangement of numbers you care to mention (or can mention); the probability being 0 for each, I suppose. The difference is, there are some infinite binary strings of numbers you cannot define without an infinite description (semantic paradoxs aside).. which one assumes, are 'truly' random. From: Norman Samish [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: The infinite list of random numbers Date: Thu, 08 Nov 2001 20:41:30 -0800 Suppose an ideal random number generator produces, every microsecond, either a zero or a one and records it on a tape. After a long time interval one would expect the tape to contain a random mix of zeroes and ones with the number of zeroes equal to the number of ones. Is this necessarily true? Is it possible that, even after an infinite time had passed, that the tape could contain all zeroes or all ones? Or MUST the tape contain an equal number of zeroes and ones? Why? If you have a reference dealing with this topic, please let me know. Thanks, Norm Samish _ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp
Re: The infinite list of random numbers
It's undefinable. You're just as likely to get all zeros, or all ones, as you are to get any arrangement of numbers you care to mention (or can mention); the probability being 0 for each, I suppose. The difference is, there are some infinite binary strings of numbers you cannot define without an infinite description (semantic paradoxs aside).. which one assumes, are 'truly' random. From: Norman Samish [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: The infinite list of random numbers Date: Thu, 08 Nov 2001 20:41:30 -0800 Suppose an ideal random number generator produces, every microsecond, either a zero or a one and records it on a tape. After a long time interval one would expect the tape to contain a random mix of zeroes and ones with the number of zeroes equal to the number of ones. Is this necessarily true? Is it possible that, even after an infinite time had passed, that the tape could contain all zeroes or all ones? Or MUST the tape contain an equal number of zeroes and ones? Why? If you have a reference dealing with this topic, please let me know. Thanks, Norm Samish _ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp
Re: The infinite list of random numbers
Yes I suppose so, there are an infinite number of ways to arrange an infinite number of zeros (or ones), but it's little odds, because they are essentially the same string as far as we are concerned. Each infinite arrangement with zeros and ones together is distinct however. More generally, all the definable arrangements of zeros and ones, would have prob. 0. From: Saibal Mitra [EMAIL PROTECTED] To: everything [EMAIL PROTECTED] Subject: Re: The infinite list of random numbers Date: Fri, 9 Nov 2001 18:45:15 +0100 All arrangemets are equally likely, but the probability is, of course, zero. So with probability one you don't get only zeros. There is a theorem that says that any finite arbitrary configuration will appear an infinite number of times in an infinite random sequence with probability one. Saibal Neil Lion wrote: It's undefinable. You're just as likely to get all zeros, or all ones, as you are to get any arrangement of numbers you care to mention (or can mention); the probability being 0 for each, I suppose. The difference is, there are some infinite binary strings of numbers you cannot define without an infinite description (semantic paradoxs aside).. which one assumes, are 'truly' random. From: Norman Samish [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: The infinite list of random numbers Date: Thu, 08 Nov 2001 20:41:30 -0800 Suppose an ideal random number generator produces, every microsecond, either a zero or a one and records it on a tape. After a long time interval one would expect the tape to contain a random mix of zeroes and ones with the number of zeroes equal to the number of ones. Is this necessarily true? Is it possible that, even after an infinite time had passed, that the tape could contain all zeroes or all ones? Or MUST the tape contain an equal number of zeroes and ones? Why? If you have a reference dealing with this topic, please let me know. Thanks, Norm Samish _ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp _ Get your FREE download of MSN Explorer at http://explorer.msn.com/intl.asp
