Comments on: A brief history of problems that ruined my world http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/ { on programming and the internets, every monday } Fri, 03 Feb 2012 14:14:15 +0000 hourly 1 http://wordpress.org/?v=3.0.1 By: tomas http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-4/#comment-28295 tomas Fri, 17 Jun 2011 23:05:32 +0000 http://lbrandy.com/blog/?p=923#comment-28295 and about the 0.9999... You can define real numbers as infinite sum for example 0.124 = 1*10^-1 + 2*10^-2 + 4+10^4 but for simplicity we just write 0.124. more general a is from 0 to 1 a = a1*10^-1 + a2*10^-2 + ... where a1,a2.. are numbers 0..9 and we simply write 0,a1a2a3... So if you write 0.999.. You should understand it as 9*10^-1 + 9*10^-2 + ... and this sum converges to one. as you probably know infinite sum 1/2 + 1/4 + 1/8 + ... + 1/2^n + ... = 1 if you write 1/2, 1/4, 1/8 in binary you get 0.1, 0.01, 0.001 etc. so 0.11111.. = 1 in binary and about the 0.9999…
You can define real numbers as infinite sum
for example

0.124 = 1*10^-1 + 2*10^-2 + 4+10^4

but for simplicity we just write 0.124.

more general a is from 0 to 1

a = a1*10^-1 + a2*10^-2 + …

where a1,a2.. are numbers 0..9 and we simply write 0,a1a2a3…
So if you write 0.999.. You should understand it as 9*10^-1 + 9*10^-2 + … and this sum converges to one.

as you probably know infinite sum

1/2 + 1/4 + 1/8 + … + 1/2^n + … = 1

if you write 1/2, 1/4, 1/8 in binary you get 0.1, 0.01, 0.001 etc.

so 0.11111.. = 1 in binary

]]> By: tomas http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-4/#comment-28294 tomas Fri, 17 Jun 2011 22:52:35 +0000 http://lbrandy.com/blog/?p=923#comment-28294 The envelope problem... I have a little problem with uniform distribution of natural numbers. But I have a similar problem. Unfortunately you have to know a little bit about set theory and what continuum hypothesis is. Ok. lets suppose a game. There are two players and target( a square [0,1]x[0,1]) they are shooting at. Lets suppose continuum hypothesis and map the target(a continuum) to the alef one(the first uncountable ordinal,a set of all countable ordinals). And and who hits the higher ordinal wins. So... Lets the first player begin. He hits a point it the target which corresponds to some ordinal, a countable ordinal!!!. So there are only countable many ordinals smaller than that ordinal the first player hit. Countable set has measure zero. So the probability that second player hits smaller ordinal than the first player is zero. So the probability that the second player wins is ONE!!! Isn't it weird that the second player always wins?? Now lets make it even weirder. Lets players one and two shoot at the same time. And figure out who has higher probability of winning? The envelope problem… I have a little problem with uniform distribution of natural numbers. But I have a similar problem. Unfortunately you have to know a little bit about set theory and what continuum hypothesis is.

Ok. lets suppose a game. There are two players and target( a square [0,1]x[0,1]) they are shooting at.
Lets suppose continuum hypothesis and map the target(a continuum) to the alef one(the first uncountable ordinal,a set of all countable ordinals). And and who hits the higher ordinal wins. So…
Lets the first player begin. He hits a point it the target which corresponds to some ordinal, a countable ordinal!!!. So there are only countable many ordinals smaller than that ordinal the first player hit. Countable set has measure zero. So the probability that second player hits smaller ordinal than the first player is zero.
So the probability that the second player wins is ONE!!! Isn’t it weird that the second player always wins??
Now lets make it even weirder. Lets players one and two shoot at the same time. And figure out who has higher probability of winning?

]]> By: JUAN PADAN http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-4/#comment-28292 JUAN PADAN Fri, 27 May 2011 16:55:23 +0000 http://lbrandy.com/blog/?p=923#comment-28292 I don't understand why thy are so difficult for you but it's intresting I don’t understand why thy are so difficult for you but it’s intresting

]]> By: Anthony http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-4/#comment-28291 Anthony Wed, 25 May 2011 01:32:43 +0000 http://lbrandy.com/blog/?p=923#comment-28291 Airplane problem: meet Mythbusters. Game over. Airplane problem: meet Mythbusters.

Game over.

]]> By: David Jameson http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-4/#comment-28289 David Jameson Mon, 23 May 2011 13:22:24 +0000 http://lbrandy.com/blog/?p=923#comment-28289 Regarding the plane problem --- stop with the "engines pushing on the air" thinking. The engines are emitting "mass" (the compressed jet) and as a result the airplane has a force pushing it in the opposite direction. As long as the wheels are allowed to spin, there is nothing to stop that force from moving the plane. Regarding the plane problem — stop with the “engines pushing on the air” thinking. The engines are emitting “mass” (the compressed jet) and as a result the airplane has a force pushing it in the opposite direction.

As long as the wheels are allowed to spin, there is nothing to stop that force from moving the plane.

]]> By: Anonymous http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-4/#comment-28283 Anonymous Tue, 17 May 2011 16:34:25 +0000 http://lbrandy.com/blog/?p=923#comment-28283 The envelope problem is simple when you approach it from a 100% rational standpoint. True, once you claim an envelope, the other envelope's expected value goes up. However, we know for a fact that this is solely the *expected* value. As we know from statistics, on a distribution of infinitely many values (due to infinitely many distinct numbers, which is shown here with the existence of an unknown variable of any potential value) any given value has a infinitely *small* probability of occurring. The envelope chosen could hold anything from $.02 to an infinitely high amount of money ($.02 is used in this case to allow for the other envelope to hold only half the amount of money of the envelope chosen.) Knowing that any specific value is almost infinitely unlikely to occur, we cannot assume any specific value of an increase in the expected value of the other envelope, forcing us to believe that (when the chosen envelope's expected value is A and the other envelopes expected value is B) B = 1.25A. True, envelope B has a higher expected value than envelope A, switching to envelope B will NOT change the TRUE value of envelope A, only the expected value. Rather than repeatedly swapping between envelopes indefinitely to raise the expected value of the other envelope (which in itself is irrational believing that the true value of an envelope would change along with the expected value), one should simply take only one envelope and accept the money, and not reason that swapping envelopes will affect the amount of money an envelope holds in any way. In short, Swapping envelopes does not increase the money held in the envelope, only the expected amount of money in the other envelope. Statistically speaking, the odds of the expected value occurring is 1/infinitely many values which in its essence, is 0. Pattern of envelopes when swapping (A, B, A, B... Pattern of expected envelope's value (A, 1.25A, 1.563A, 1.953A... Pattern of actual envelope's value (A, 2A/.5A, A, 2A/.5A... While the expected value DOES increase, the actual value remains more or less constant. The envelope problem is simple when you approach it from a 100% rational standpoint. True, once you claim an envelope, the other envelope’s expected value goes up. However, we know for a fact that this is solely the *expected* value. As we know from statistics, on a distribution of infinitely many values (due to infinitely many distinct numbers, which is shown here with the existence of an unknown variable of any potential value) any given value has a infinitely *small* probability of occurring. The envelope chosen could hold anything from $.02 to an infinitely high amount of money ($.02 is used in this case to allow for the other envelope to hold only half the amount of money of the envelope chosen.) Knowing that any specific value is almost infinitely unlikely to occur, we cannot assume any specific value of an increase in the expected value of the other envelope, forcing us to believe that (when the chosen envelope’s expected value is A and the other envelopes expected value is B) B = 1.25A. True, envelope B has a higher expected value than envelope A, switching to envelope B will NOT change the TRUE value of envelope A, only the expected value. Rather than repeatedly swapping between envelopes indefinitely to raise the expected value of the other envelope (which in itself is irrational believing that the true value of an envelope would change along with the expected value), one should simply take only one envelope and accept the money, and not reason that swapping envelopes will affect the amount of money an envelope holds in any way.

In short, Swapping envelopes does not increase the money held in the envelope, only the expected amount of money in the other envelope. Statistically speaking, the odds of the expected value occurring is 1/infinitely many values which in its essence, is 0.

Pattern of envelopes when swapping (A, B, A, B…
Pattern of expected envelope’s value (A, 1.25A, 1.563A, 1.953A…
Pattern of actual envelope’s value (A, 2A/.5A, A, 2A/.5A…

While the expected value DOES increase, the actual value remains more or less constant.

]]> By: Matt http://lbrandy.com/blog/2009/07/a-brief-history-of-problems-that-ruined-my-world/comment-page-3/#comment-28282 Matt Tue, 17 May 2011 01:48:37 +0000 http://lbrandy.com/blog/?p=923#comment-28282 Umm, The airplane would definitely not take off, the force that is used to get the plane in the air is provided by the shape of the wing. It is designed in such a way that the air that goes over the top of the wing has to travel a farther distance to get around the wing and consequently, has to travel faster. When air is traveling fast, it has less pressure than slower air around it. Considering how air moves from areas of high pressure to low pressure (AKA wind), The air that is under the wing moves up to compensate for the lower pressure above the wing. That is what gives the plane its lift. The engine has nothing to do with it at all, the only thing the engine does is move the plane forward to provide the air traveling across the wing, and of course so that it can go! :P Umm, The airplane would definitely not take off, the force that is used to get the plane in the air is provided by the shape of the wing. It is designed in such a way that the air that goes over the top of the wing has to travel a farther distance to get around the wing and consequently, has to travel faster. When air is traveling fast, it has less pressure than slower air around it. Considering how air moves from areas of high pressure to low pressure (AKA wind), The air that is under the wing moves up to compensate for the lower pressure above the wing. That is what gives the plane its lift. The engine has nothing to do with it at all, the only thing the engine does is move the plane forward to provide the air traveling across the wing, and of course so that it can go! :P

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