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Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 01:53:00 - [1] Case 1D : infinite line of pixels, start at coordinate (0) Case 2D : infinite plane of pixels, start at coordinate (0,0) Case 3D : infinite volume of pixels, start at coordinate (0,0,0) In each of the cases, you move your cursor one pixel in a random direction - equal probabilities of moving in any of the possible directions. For 1D, you can move "left" or "right" with 50% probability for each (1/2). For 2D, you can move "left", "right", "up" or "down" with 25% probability for each (1/4). For 3D, you can move "left", "right", "up", "down", "forwards" or "backwards" with 16.66(6)% probability for each (1/6). GIVEN AN INFINITE AMOUNT OF TIME, what is the probability you will EVENTUALLY land on the origin point at least once again for each of the three cases ? _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Calathea Sata State War Academy	Posted - 2011.04.16 02:06:00 - [2] Probability? 100%
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 02:24:00 - [3] Originally by: Calathea Sata Probability? 100% For which one(s) ? _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Epegi Givo Amarr Department of Redundancy Dpt.	Posted - 2011.04.16 02:30:00 - [4] Given an infinite amount of time for each one, shouldn't the probability for each case be 1.0? They asked me if I was a God, and I said no, because God doesn't have an annoying sig.
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 02:34:00 - [5] Edited by: Akita T on 16/04/2011 02:34:41 Originally by: Epegi Givo Given an infinite amount of time for each one, shouldn't the probability for each case be 1.0? I don't know, is it ? _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Dray Caldari Euphoria Released Merciless.	Posted - 2011.04.16 03:48:00 - [6] Statistics and infinity in the same puzzle, I'd rather you just trolled me. Given infinity the chances of landing anywhere not just your start point would be 100%? It's like 5.00am and I've got 12 hours of work ahead of me and my head is f**ked already.
Culmen Caldari Blood Phage Syndicate Narwhals Ate My Duck	Posted - 2011.04.16 03:49:00 - [7] Originally by: Akita T Edited by: Akita T on 16/04/2011 02:41:50 Originally by: Epegi Givo Given an infinite amount of time for each one, shouldn't the probability for each case be 1.0? I don't know, is it ? Care to quickly explain the reasoning for each individual case ? I a bit busy for a formal proof. But here's a rough sketch i haven't verified. The odds of a length 2 move returning are 50 % the odds of a length 2+ are greater(?) than 50%, because they include the odds of it reaching in 2 moves Infinite Summation of 1/2 = 1. Of course any rigourous mathematical analysis is going to blow this apart, but I am kinda drunk right now. and further more why do i even need a sig?
Calathea Sata State War Academy	Posted - 2011.04.16 03:55:00 - [8] Since the probabilities covers all directions, given an infinite amount of time to move, all possible positions will be eventually landed upon. Probability of any of the positions to be landed upon eventually is 100% because it is an eventual outcome not a probable outcome.
Sitara Minmatar Solar Flare Trade and Production	Posted - 2011.04.16 03:57:00 - [9] Ok - here goes. With an infinite amount of time you will visit each point that can be reached an infinite number of times. So all we need to show is that the origin can be reached again from any point and the probability is 100% In case 1D - trivial - 2 possible directions to move at each step and you can take either - say you're at position x where x!=0 then if x<0 move x+1 else move x-1 ; repeat until back at origin : 100% In case 2D - extension of above. 4 possible directions. if at any point (x,y) apply the rule above to each of x and y till you reach the origin : 100% In case 3D - extension of above. 6 possible directions. if at point (x, y, z) apply the rule to each of x, y and z till you reach the origin : 100%
Sturmwolke	Posted - 2011.04.16 04:01:00 - [10] What do you mean exactly by land on the origin point? They all already start at zero .. unless you plan to drop them off somewhere random. Even if these masses of pixels become sentient and wanders off forming colonies somewhere, the factory still makes them at "zero".
Calathea Sata State War Academy	Posted - 2011.04.16 04:02:00 - [11] OMG WAIT - ITS A TRAP It didn't mention how long it takes to move the pixel. It only said "you can move..." so you can move only once in all cases. So the probability of landing at the origin is 0% because there is no probability for "staying still".
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 04:43:00 - [12] Edited by: Akita T on 16/04/2011 04:47:58 HINTS : You can only ever get back to the origin in an even step (2, 4, 6, 8, etc) because you need the number of moves in any one direction to be equal to that in the exact opposite direction, regardless of example. For 1D, this is the simplest case. The probability to get back to coordinate (0) in EXACTLY the "2x"th step is equal to the probability of rolling x "lefts" and x "rights" precisely, or combinations(2x,x) {= 2x!/(x!x!)* divided over number of possible combinations in 2x steps, namely 2^2x. So, for step 2 (x=1), you get (2!/1!)/2^2 = 2/4 = 50% chance of return (left+right or right+left, as opposed to left+left or right+right, 2 out of 4), same 50% chance of no return. For step 4 (x=2), you get (4!/2!^2)/2^4 = 6/16 = 37.5% chance of return, 62.5% chance of no return, a 0.6250.5= 0.3125 chance of no return at all, so a 68.75% cumulative chance of at least a return so far. For step 6 (x=3), you get (6!/3!^2)/2^6 = (720/36)/64 = 20/64 = 31.25% chance of return, 68.75 chance of no return, a 0.68750.3125=0.21484375 chance of no return at all, so a cumulative 78.515625% chance of at least a return so far. Now you have to prove that this specific series tends towards 100% when x tends to infinity. For the "1D" case, it certainly does, and that's actually the easy part to prove. 2D is more complicated. 3D, even worse. UBERHINT : The answer is NOT the same for all three cases. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:02:00 - [13] It is one. The chance that you land on the origin is as high as landing on any point, and because you have an infinite amount of time and a (countable) infinite amount of positions will you get to any point eventually. --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:04:00 - [14] Originally by: Whitehound It is one. The chance that you land on the origin is as high as landing on any point, and because you have an infinite amount of time and a (countable) infinite amount of positions will you get to any point eventually. In the first case ? Absolutely ! But are you sure about the second case ? What about the last case ? _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:05:00 - [15] Edited by: Whitehound on 16/04/2011 05:08:05 Originally by: Akita T Originally by: Whitehound It is one. The chance that you land on the origin is as high as landing on any point, and because you have an infinite amount of time and a (countable) infinite amount of positions will you get to any point eventually. In the first case ? Absolutely ! But are you sure about the second case ? What about the last case ? Just as countable infinite. Look up Cantor's diagonal method (or argument). --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:09:00 - [16] Edited by: Akita T on 16/04/2011 05:14:40 ~~@Whitehound : How much ISK are you willing to bet (using a trusted 3rd party) on that answer ?~~ Well, let's just say that there is no completely correct answer so far. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:14:00 - [17] Edited by: Whitehound on 16/04/2011 05:14:57 Originally by: Akita T @Whitehound : How much ISK are you willing to bet (using a trusted 3rd party) on that answer ? Are you asking for help or just to fool people? If it is the later then you are only having wasted my time, girl. --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:15:00 - [18] I already know the exact answers for each of the three cases, including proofs. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:18:00 - [19] Originally by: Akita T I already know the exact answers for each of the three cases, including proofs. Here is a new one: what are the chances of you being wrong? --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:19:00 - [20] Edited by: Akita T on 16/04/2011 05:28:06 Originally by: Whitehound Here is a new one: what are the chances of you being wrong? I also USED to mistakenly believe the correct answer is "100% for ALL three cases", but it's not. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:29:00 - [21] Edited by: Whitehound on 16/04/2011 05:32:02 Originally by: Akita T Originally by: Whitehound Here is a new one: what are the chances of you being wrong? Practically zero for this particular set of problems. I ALSO used to mistakenly believe the correct answer is "100% for all three cases". It is the same for all three cases and you can use Cantor's diagonal argument to create an infinite 1-dimensional space out of any finite-dimensional space that consists out of countable infinite elements. The amount of movements you do is also finite, and I assume an equally distributed randomness, so I do not see where you think there is a difference between having 1, 2 or 3 dimensions. You have also not given any further characteristics that would allow for an answer other than 0 or 1, and it cannot be zero as long as one has got infinite time. You will eventually get back to the point. If this takes you 1infinite time, or 4infinite time, or 1,000,000*infinite time does not matter, it just stays infinite. So if you think it is different then it is you who is wrong. --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:32:00 - [22] Edited by: Akita T on 16/04/2011 05:35:12 If you promise to keep it secret so the rest can also wrack their brains on it a little more, I can evemail you the PROOF that you are wrong. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:34:00 - [23] Edited by: Whitehound on 16/04/2011 05:36:57 Originally by: Akita T If you promise to keep it secret so the rest can also wrack their brains on it a little more, I can evemail you the PROOF that you are wrong. I am not interested, Akita. I know I am right and that you are wrong. If you think you are right then check if your initial question corresponds to your answer. You might want to check on this part first: "equal probabilities of moving in any of the possible directions in each move." This can mean anything, really. --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:38:00 - [24] Originally by: Whitehound I know I am right and that you are wrong. If you think you are right then check if your initial question corresponds to your answer. Like I said, I also used to think I was right, but I was wrong, the proof is rock solid, and the initial question is phrased correctly. It's just one of those cases where gut feeling and math clash, and the actual answer is quite surprising. Check back later after several others have given it a try too, to see just why you were wrong. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:39:00 - [25] Originally by: Akita T Check back later after several others have given it a try too, to see just why you were wrong. No. I am not wrong, but you are. --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:42:00 - [26] Edited by: Akita T on 16/04/2011 05:42:51 Originally by: Whitehound No. I am not wrong, but you are. Dude, you're going to feel so uncomfortable when you see the actual answers... Seriously, it's not so easy as it looks at first sight. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:42:00 - [27] Originally by: Akita T It means exactly what it says. It means BS, Akita. You would not be the first student who fell for it. --
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:44:00 - [28] Originally by: Akita T Originally by: Whitehound No. I am not wrong, but you are. Dude, you're going to feel so uncomfortable when you see the actual answers... No, I will not. You will only reveal more of what you were truly asking. You only do not know it yet. --
Akita T Caldari Navy Volunteer Task Force	Posted - 2011.04.16 05:49:00 - [29] Edited by: Akita T on 16/04/2011 05:51:08 Originally by: Whitehound Originally by: Akita T Originally by: Whitehound No. I am not wrong, but you are. Dude, you're going to feel so uncomfortable when you see the actual answers... No, I will not. You will only reveal more of what you were truly asking. You only do not know it yet. It's a trap question, but not in the way you seemed to think it might be. I can assure you that the wording of the problem is exactly as it should be, the first case is trivial, the last case is horrible. There is absolutely no trickery with regards to the wording, it's just that "common sense" itself plays tricks on you when you try to solve it quickly. Like I said, a LOT of people get this wrong, there's no shame in getting it wrong by yourself. _ CCP LEADERSHIP MENTALITY NEEDS TO CHANGE FAST ! "New junky features sell, old polished content doesn't" ? KILL IT WITH FIRE.
Whitehound The Whitehound Corporation Frontline Assembly Point	Posted - 2011.04.16 05:54:00 - [30] Originally by: Akita T It's a trap question, but not in the way you seemed to think it might be. I can assure you that the wording of the problem is exactly as it should be, the first case is trivial, the last case is horrible. There is absolutely no trickery with regards to the wording, it's just that "common sense" itself plays tricks on you when you try to solve it quickly. Like I said, a LOT of people get this wrong, there's no shame in getting it wrong by yourself. Who cares? If it is a trap question then you have confirmed the BS. What else is there to know? No one gives a crap about your BS, Akita. --

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