Stathis, Thank you for the explanation. Yes, it is indeed surprising to discover that if something CAN happen, it may NOT happen, even in infinite time - provided that the chance of it happening decreases with time.
Nevertheless, the proverbial monkey at the typewriter, randomly hitting the keys, must eventually write all the works of Shakespeare, as well as everything that has been written or will be written, provided he spends eternity at the task. As you suggested, I used the spreadsheet to determine the product of (1-1/4)(1-1/9)(1-1/16)...(1-1/n^2) with n going from 1 through 65,536 and got 0.500,007,629,510,935. Extending n to 131,072 resulted in 0.500,007,596,291,283. It looks like n would have to get to a large number before the product would be 0.5, but I'm now convinced that it would eventually get there. It is counter-intuitive to me that the spreadsheet sum of 1/n^2, with n going from 1 through 131,072 , is 0.644,918,874,381,156 rather than 0.5. We live in a marvelous universe! Norman ----- Original Message ----- From: "Stathis Papaioannou" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Wednesday, July 21, 2004 10:11 PM Subject: RE: Math Problem Norman, Perhaps the term "cumulative probability" was misleading. I meant the probability that P will occur at least once between t=2 and t=infinity. Suppose you enter a strange lottery in its second week, when the probability P that you will win is Pr(P)=1/(t^2)=1/(2^2)=1/4. The lottery runs every week from now until eternity, but your chance of winning continuously falls so that after t weeks Pr(P)=1/(t^2); that is, in the third week you have a 1/9 chance of winning, in the fourth week 1/16 chance, and so on. Note that Pr(P)>0 for all integers t, no matter how large. Now, the question is if you play this lottery forever, what is the probability that you will win at least once? (This is what I meant by cumulative probability.) It is easier mathematically to rephrase this question as, what is the chance that you will NEVER win (Pr(~P)) after t consecutive weeks as t->infinity? Since Pr(P)=1/(t^2), Pr(~P)=1-1/(t^2). So the infinite product (1-1/4)(1-1/9)(1-1/16)... gives the probability that you will play this game forever and never win. If you multiply out the above sequence, you will see that it limits to 1/2. What this means is that there are events which are possible at all times, but nevertheless will NEVER occur, even given an eternity. This is what I found surprising. So whether a particular event does or doesn't actually occur in the future of the universe depends less on the absolute probability (if we have infinite time) than on how the probability varies as a function of time. I have not posted this to the list as you only posted your question to me; if you think this reply would be of interest to the list, please feel free to forward it. Stathis Papaioannou >From: "Norman Samish" <[EMAIL PROTECTED]> >To: "Stathis Papaioannou" <[EMAIL PROTECTED]> >Subject: Math Problem >Date: Wed, 21 Jul 2004 10:36:13 -0700 > >Stathis Papaioannou, > >I don't understand your statement "For example, if Pr(P)=1/(t^2), as t goes >from 2 to infinity, the cumulative probability that P will occur at some >point is 1/2." At first glance this looked correct, but when I ran out >Pr(P)=1/(t^2) on a spreadsheet, with t going in steps of 1 from 2 to 65536, >the cumulative probability seems to go to 0.644019, not 1/2. > >The cumulative probability seems to depend on the step size. If t goes in >steps of 1.448, then the cumulative probability goes to 1/2. By other >choices of step size, I can make the cumulative probability sum to anything >between 1/4 and 16383. > >An explanation will be appreciated. Thanks, > >Norman Samish > >----- Original Message ----- >Subject: Re: All possible worlds in a single world cosmology? > > >At 23:12 21/07/04 +1000, Stathis Papaioannou wrote: > > > The increase in entropy and cooling which go with the model I suggested > > are average trends over time. It is possible within this long term > > decline to have pockets of order/ decreasing entropy, both in classical > > statistical mechanics and quantum mechanics. It is a mathematical fact, > > independent of the actual physics, that given enough time (and eternity > > is certainly enough time), any event that is possible, however close to > > zero its probability per unit time, will occur with probability > > arbitrarily close to 1. What rather surprised me, however, is the fact > > that the last statement is only true in general if the probability per > > unit time stays constant or increases with increasing time; if it > > decreases, limiting towards zero as time approaches infinity, then it is > > possible that this event, which still always has non-zero probability >per > > unit time, may never actually occur. For example, if Pr(P)=1/(t^2), as t > > goes from 2 to infinity, the cumulative probability that P will occur at > > some point is 1/2.