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Mathematicians shocked to find pattern in "random" prime numbers

Topic closed. 40 replies. Last post 9 months ago by amber123.

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Posted: March 15, 2016, 9:07 pm - IP Logged

I was reading another article on the same subject https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/ and came across this interesting tidbit --

Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

 

Really? Why would that be true? 

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    Posted: March 16, 2016, 7:03 am - IP Logged

    I was reading another article on the same subject https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/ and came across this interesting tidbit --

    Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

     

    Really? Why would that be true? 

    It's one of those counter intuitive logic problems that make no sense at first, but when you do the math it makes every sense. Like the Monty Hall problem or the birthday problem.

    The solution here is quite simple;

    If Alice fails to get a tail after a head, she only needs to toss again and keep doing it until she gets a tail. Bob on the other hand, if he doesn't get a head after his first head, needs to start over again. Alice's trials don't reset after a failure, whereas Bob's trials do. So on average he will take longer.

      Luminus's avatar - ouskuu

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      Posted: March 16, 2016, 11:50 am - IP Logged

      I was reading another article on the same subject https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/ and came across this interesting tidbit --

      Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

       

      Really? Why would that be true? 

      Actually, the head and tail don't have an equal chance of appearing.  The head side is slightly heavier, which is why it tends to land with that side up more often.

       

      http://www.smithsonianmag.com/science-nature/gamblers-take-note-the-odds-in-a-coin-flip-arent-quite-5050-145465423/?no-ist

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        Posted: March 16, 2016, 12:01 pm - IP Logged

        Actually, the head and tail don't have an equal chance of appearing.  The head side is slightly heavier, which is why it tends to land with that side up more often.

         

        http://www.smithsonianmag.com/science-nature/gamblers-take-note-the-odds-in-a-coin-flip-arent-quite-5050-145465423/?no-ist

        That's only an added effect. Even if we had a fair coin whose both sides weighted exactly the same, Alice would still be winning. Even if we said that Bob needs two tails instead of two heads, with a fair coin, Alice would still be winning.

          Luminus's avatar - ouskuu

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          Posted: March 17, 2016, 12:01 pm - IP Logged

          That's only an added effect. Even if we had a fair coin whose both sides weighted exactly the same, Alice would still be winning. Even if we said that Bob needs two tails instead of two heads, with a fair coin, Alice would still be winning.

          How do you know?  Has this been tested?  I can't see why that would be, since the head and tail are just engraved on the coin.  Technically, both sides are equal, meaning there is no head or tail.

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            Posted: March 17, 2016, 4:19 pm - IP Logged

            How do you know?  Has this been tested?  I can't see why that would be, since the head and tail are just engraved on the coin.  Technically, both sides are equal, meaning there is no head or tail.

            Yes, I just wrote an application to test it and am getting 4 for Alice and 6 for Bob on average. I will post it here as soon as my no links restriction is lifted. You can also test it yourself by simply tossing a coin, but it might take a while to get a large enough sample to calculate averages.

            But you don't really need to test to know this. As explained earlier, Alice's trials don't reset whereas Bob's do. It has nothing to do with coin's physical characteristics.

              rundown99's avatar - cigar

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              Posted: March 18, 2016, 11:48 am - IP Logged

              The truth is that the numbers are ONLY used to identify the balls and distinguish the balls from one another.  No mathematical formula is going to predict the Powerball lottery game.  For example, the balls selected from Wednesday's Powerball drawing were 10, 12, 13, 46, 50 and red ball 21.  if you picked 9, 11, 14, 45, 49, and red ball 20, you were no closer to the jackpot that if someone had picked 1,2,3,4,5 and Powerball 10.  You either matched those balls or you didn't.

              Smart lottery winners form trust to claim their winnings.  They send an attorney to the lottery headquarters to claim the prize in trust, so that ONLY the name of the trust is revealed.  And they tell NO ONE, especially relatives.

              If you ever win a lottery and you are single, the only person you should ever marry is someone who was truly in love with you BEFORE you won the jackpot!

                Luminus's avatar - ouskuu

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                Posted: March 18, 2016, 12:14 pm - IP Logged

                The truth is that the numbers are ONLY used to identify the balls and distinguish the balls from one another.  No mathematical formula is going to predict the Powerball lottery game.  For example, the balls selected from Wednesday's Powerball drawing were 10, 12, 13, 46, 50 and red ball 21.  if you picked 9, 11, 14, 45, 49, and red ball 20, you were no closer to the jackpot that if someone had picked 1,2,3,4,5 and Powerball 10.  You either matched those balls or you didn't.

                Maybe or maybe not.  I read about a mathematician who claims that you can predict lottery numbers more likely to be drawn:

                 

                http://www.dailymail.co.uk/sciencetech/article-2585157/You-CAN-predict-lottery-numbers-Brazilian-mathematician-claims.html

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                  Posted: March 18, 2016, 1:10 pm - IP Logged

                  Maybe or maybe not.  I read about a mathematician who claims that you can predict lottery numbers more likely to be drawn:

                   

                  http://www.dailymail.co.uk/sciencetech/article-2585157/You-CAN-predict-lottery-numbers-Brazilian-mathematician-claims.html

                  That's a crackpot. Explained some time ago by Karl Lietchy.

                    sully16's avatar - sharan
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                    Posted: March 18, 2016, 1:10 pm - IP Logged

                    The Devine number 1.618

                    Did you exchange a walk on part in the war ?

                    For a lead role in a cage?

                     

                                                                From Pink Floyd's " Wish you were here"

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                      Posted: March 18, 2016, 6:17 pm - IP Logged

                      How do you know?  Has this been tested?  I can't see why that would be, since the head and tail are just engraved on the coin.  Technically, both sides are equal, meaning there is no head or tail.

                      Testing will confirm it, but simple logic explains why it has to be true.

                      Both Bob and Alice need to start by getting the first  heads, so that part will be exactly equal. They also both have the same 50% chance of success on the next flip. It's what happens the 50% of the time that they fail on the next flip that results in the difference.

                      When Alice fails to get a tails she already has the first heads that she needs in order to get a heads followed by a tails. That means she can succeed with one flip after the failure: first H > failing H >  successful T

                      When Bob fails to get his consecutive heads he has a tail, and therefore has to flip at least once more to get the first heads, and then flip again to get a second heads: first H > Failing T > first H > successful H

                      This is actually somewhat similar to the discovery about prime numbers. It's not the nature of prime numbers or the equal probability of H/T that really matters. It's the sequence from one event to the next.

                        Luminus's avatar - ouskuu

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                        Posted: March 19, 2016, 8:07 am - IP Logged

                        That's a crackpot. Explained some time ago by Karl Lietchy.

                        Oh well.LOL

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                          Posted: March 19, 2016, 11:42 am - IP Logged

                          In case anyone wants to simulate this coin toss "paradox" you can download the application from here. Just run the CoinTossParadoxSimulator.exe. If it doesn't work, run the setup.exe first.

                          You may need to make exclusions in your antivirus software. In case you need to run the setup.exe, if the setup finishes with an error, don't worry It's just an antivirus quirk.


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                            Posted: March 20, 2016, 10:43 am - IP Logged

                            I'm a maths moron, but I think you guys are over-thinking (at least when it comes to the Pick3 games). After reading this thread, I set out to find a fresh example of a simple pattern in the numbers and it only took me two days to find one. I publicly called last night's Pennsylvania P3 result using software that pulls the world's simplest pattern out of the numbers. It looks like this: 

                            02/26/2016 064;236 | 04/08/2014 064;182 | 09/07/2012 098;064
                            02/27/2016 337;315 | 04/09/2014 196;528 | 09/08/2012 602;260
                            02/28/2016 430;637 | 04/10/2014 590;768 | 09/09/2012 273;378
                            02/29/2016 454;545 | 04/11/2014 716;008 | 09/10/2012 801;046
                            03/01/2016 102;574 | 04/12/2014 601;601 | 09/11/2012 497;671
                            03/02/2016 610;852 | 04/13/2014 479;866 | 09/12/2012 953;573
                            03/03/2016 416;402 | 04/14/2014 317;287 | 09/13/2012 471;205
                            03/04/2016 460;276 | 04/15/2014 630;368 | 09/14/2012 991;705
                            03/05/2016 857;939 | 04/16/2014 207;759 | 09/15/2012 857;985
                            03/06/2016 703;125 | 04/17/2014 702;298 | 09/16/2012 140;342
                            03/07/2016 834;756 | 04/18/2014 476;969 | 09/17/2012 595;392
                            03/08/2016 246;551 | 04/19/2014 722;441 | 09/18/2012 113;497
                            03/09/2016 033;857 | 04/20/2014 040;579 | 09/19/2012 962;016
                            03/10/2016 913;410 | 04/21/2014 688;244 | 09/20/2012 739;865
                            03/11/2016 884;713 | 04/22/2014 712;773 | 09/21/2012 943;163
                            03/12/2016 762;960 | 04/23/2014 618;036 | 09/22/2012 689;364
                            03/13/2016 281;485 | 04/24/2014 345;184 | 09/23/2012 607;340
                            03/14/2016 729;311 | 04/25/2014 930;513 | 09/24/2012 149;773
                            03/15/2016 645;679 | 04/26/2014 367;173 | 09/25/2012 853;489
                            03/16/2016 372;386 | 04/27/2014 458;979 | 09/26/2012 057;629
                            03/17/2016 911;601 | 04/28/2014 419;565 | 09/27/2012 250;736
                            03/18/2016 036;052 | 04/29/2014 141;547 | 09/28/2012 942;360
                            03/19/2016 301;070 | 04/30/2014 070;028 | 09/29/2012 070;057

                            All you do is look for number pairings at a certain number of days. It's not hard, just create a dictionary of every number pair within a window of days (I arbitrarily use 30 to 40, we tried up to 1000 days without better results), then sort the dictionary on the first number + the second number + the number of days, then accumulate a list of the pairings that have happened more than once and reconcile it with the numbers which have fallen most recently within the window you chose to begin with. 

                            I know I'm a knuckle-dragging, mouth-breathing maths moron, but you have to admit, it's a little peculiar how easy it was for me to find a fresh hit based on this idiotically-simple idea. I did it prospectively to give you guys a new example to think about. I know you guys are going to quickly say, "Let's see you do it again!" To which I quickly reply, "Okay." Heh. 

                            Seriously guys, I know I'm just deluding myself, but I would sincerely appreciate it if one of you maths folks would at least take a whack at this just to make once-and-for-all sure that I'm wrong.

                              amber123's avatar - OpIFNim

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                              Posted: March 20, 2016, 11:32 am - IP Logged

                              I'm a maths moron, but I think you guys are over-thinking (at least when it comes to the Pick3 games). After reading this thread, I set out to find a fresh example of a simple pattern in the numbers and it only took me two days to find one. I publicly called last night's Pennsylvania P3 result using software that pulls the world's simplest pattern out of the numbers. It looks like this: 

                              02/26/2016 064;236 | 04/08/2014 064;182 | 09/07/2012 098;064
                              02/27/2016 337;315 | 04/09/2014 196;528 | 09/08/2012 602;260
                              02/28/2016 430;637 | 04/10/2014 590;768 | 09/09/2012 273;378
                              02/29/2016 454;545 | 04/11/2014 716;008 | 09/10/2012 801;046
                              03/01/2016 102;574 | 04/12/2014 601;601 | 09/11/2012 497;671
                              03/02/2016 610;852 | 04/13/2014 479;866 | 09/12/2012 953;573
                              03/03/2016 416;402 | 04/14/2014 317;287 | 09/13/2012 471;205
                              03/04/2016 460;276 | 04/15/2014 630;368 | 09/14/2012 991;705
                              03/05/2016 857;939 | 04/16/2014 207;759 | 09/15/2012 857;985
                              03/06/2016 703;125 | 04/17/2014 702;298 | 09/16/2012 140;342
                              03/07/2016 834;756 | 04/18/2014 476;969 | 09/17/2012 595;392
                              03/08/2016 246;551 | 04/19/2014 722;441 | 09/18/2012 113;497
                              03/09/2016 033;857 | 04/20/2014 040;579 | 09/19/2012 962;016
                              03/10/2016 913;410 | 04/21/2014 688;244 | 09/20/2012 739;865
                              03/11/2016 884;713 | 04/22/2014 712;773 | 09/21/2012 943;163
                              03/12/2016 762;960 | 04/23/2014 618;036 | 09/22/2012 689;364
                              03/13/2016 281;485 | 04/24/2014 345;184 | 09/23/2012 607;340
                              03/14/2016 729;311 | 04/25/2014 930;513 | 09/24/2012 149;773
                              03/15/2016 645;679 | 04/26/2014 367;173 | 09/25/2012 853;489
                              03/16/2016 372;386 | 04/27/2014 458;979 | 09/26/2012 057;629
                              03/17/2016 911;601 | 04/28/2014 419;565 | 09/27/2012 250;736
                              03/18/2016 036;052 | 04/29/2014 141;547 | 09/28/2012 942;360
                              03/19/2016 301;070 | 04/30/2014 070;028 | 09/29/2012 070;057

                              All you do is look for number pairings at a certain number of days. It's not hard, just create a dictionary of every number pair within a window of days (I arbitrarily use 30 to 40, we tried up to 1000 days without better results), then sort the dictionary on the first number + the second number + the number of days, then accumulate a list of the pairings that have happened more than once and reconcile it with the numbers which have fallen most recently within the window you chose to begin with. 

                              I know I'm a knuckle-dragging, mouth-breathing maths moron, but you have to admit, it's a little peculiar how easy it was for me to find a fresh hit based on this idiotically-simple idea. I did it prospectively to give you guys a new example to think about. I know you guys are going to quickly say, "Let's see you do it again!" To which I quickly reply, "Okay." Heh. 

                              Seriously guys, I know I'm just deluding myself, but I would sincerely appreciate it if one of you maths folks would at least take a whack at this just to make once-and-for-all sure that I'm wrong.

                              Wow, you got 070 straight yesterday with just two sets. That's impressive.

                              Can you do a step by step for Florida so I can understand exactly what you are doing? Thanks.