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Fooled by Randomness

Topic closed. 297 replies. Last post 6 years ago by jimmy4164.

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Posted: August 3, 2010, 6:15 pm - IP Logged

I will avoid yours, and I think others will too.

End of your thread. Bye!


    United States
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    July 10, 2010
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    Posted: August 3, 2010, 10:16 pm - IP Logged

    The Quest for a Blockbuster System

    In a couple of my earlier posts, I mentioned that I had 12,200 Draw results for the Pennsylvania Daily Number game (Evening).  This was incorrect.  I hadn't finished grabbing all 33.4 years from the PA website, so I multipled 33.4 by 365 and rounded to 12,200.  In fact, PA did not draw Daily Numbers on Sundays for nearly 2 years.  So, now that I'm done, what I have are 11,563 rows from March 1, 1977 to today, August 3, 2010.

    Theoretically, according to probability as I learned it, a particular 3 digit number is expected to be drawn, on average, once every 1000 draws, provided there is no human or mechanical interference in the selection process.  This theory can be tested easily at the PA website by merely entering the number of interest in the Daily Number (Evening) "Has my number ever hit before" boxes, and clicking Search.  Now, since there have been approximately 12,000 draws, we should expect each number to be drawn around 12 times, give or take.  The give or take is measured in standard deviations, which I have not yet calculated.  However, more often than not, I observe the number of hits to fall in the range 9 to 15 over the 33.4 year period, which looks normal.  For example, 328 had 13 hits, 493 had 11, and 473, 12.  Some results that some here will not want to believe are 111:12 hits, 888:13, and 000:14. The results are all public information at the PA Official site.

    Which brings us to some more complicated backtests, those that aren't available seamlessly at the PA site.  For starters, I just wrote a tiny program which scans the 11,563 lines in the file to answer some of the easier questions.  The first question I asked was, What would have happened if each day I had bought a straight ticket on the PA Daily Number (Evening) , and religously chose the WINNING NUMBER OF THE PREVIOUS DAY?  So, here is our first SYSTEM in the Fooled by Randomness thread!  Always buy the previous day's winning number.  (Send out a Toad Alert!)  Smile 

    Hopefully this will be the prelude to testing some of the more sophisticated Systems published here for 3 digit lotteries.

    According to the same probability theory used to calculate the number 12 above, for more or less random selections, the results for yesterday's winners should be the same!  On any given day, in a fair game, the odds of a specific number hitting is 1:1000.  Now, will this apply to a choice like "Yesterday's Winner?"  We will soon see.  But first, I'd like to satisfy my curiosity by staging a Poll here.  I've never done one, so I'm not sure where it will be seen when I initiate it from this Forum/Thread.  This site is pretty cool, so I'm sure it will work out.

    Look for the Poll soon!

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      NASHVILLE, TENN
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      Posted: August 6, 2010, 12:13 pm - IP Logged

      So, here is our first SYSTEM in the Fooled by Randomness thread!  Always buy the previous day's winning number.  (Send out a Toad Alert!)  Smile 

       

      Why should I buy the previous day's winning number?


        United States
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        Posted: August 6, 2010, 1:09 pm - IP Logged

        So, here is our first SYSTEM in the Fooled by Randomness thread!  Always buy the previous day's winning number.  (Send out a Toad Alert!)  Smile 

         

        Why should I buy the previous day's winning number?

        GASMETERGUY,

        I didn't say you should.  This is just an exercise to prove that buying yesterday's winning number Straight will result in approximately the same number of hits over 33+ years as you would expect buying QPs.  Stay tuned!

        --Jimmy4164

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          NASHVILLE, TENN
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          Posted: August 6, 2010, 8:49 pm - IP Logged

          Always buy the previous day's winning number. 

          Sorry.  My bad.  I read "always" and assumed you meant "You should".


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            Posted: August 7, 2010, 12:22 am - IP Logged

            Always buy the previous day's winning number. 

            Sorry.  My bad.  I read "always" and assumed you meant "You should".

            You're right, I should have been more clear about what that boldfaced sentence really was.  RedToad had been complaining about my lack of a System in this Forum and this was my [I guess confusing] way of pulling his chain a little.  Thanks for your interest and comments.

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              NASHVILLE, TENN
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              Posted: August 7, 2010, 10:55 am - IP Logged

              Speaking of Red Toad, IMHO he was only trying to provoke you into revealing your thoughts instead of your results.  I coult be wrong; I often am; but that is the way I read his posts.

              I, too, am only interested in your thoughts.  Results (i.e. the posting of numbers) means nothing.  I could do that for myself.  How you achieved those numbers is the main interest.  Being the kind, patient person that I am (I pulled a muscle patting myself on the back for that one) I will wait upon your pleasure.  I am sure you will reveal your thinking in your own good time.

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                Kentucky
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                February 14, 2006
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                Posted: August 7, 2010, 9:49 pm - IP Logged

                The Quest for a Blockbuster System

                In a couple of my earlier posts, I mentioned that I had 12,200 Draw results for the Pennsylvania Daily Number game (Evening).  This was incorrect.  I hadn't finished grabbing all 33.4 years from the PA website, so I multipled 33.4 by 365 and rounded to 12,200.  In fact, PA did not draw Daily Numbers on Sundays for nearly 2 years.  So, now that I'm done, what I have are 11,563 rows from March 1, 1977 to today, August 3, 2010.

                Theoretically, according to probability as I learned it, a particular 3 digit number is expected to be drawn, on average, once every 1000 draws, provided there is no human or mechanical interference in the selection process.  This theory can be tested easily at the PA website by merely entering the number of interest in the Daily Number (Evening) "Has my number ever hit before" boxes, and clicking Search.  Now, since there have been approximately 12,000 draws, we should expect each number to be drawn around 12 times, give or take.  The give or take is measured in standard deviations, which I have not yet calculated.  However, more often than not, I observe the number of hits to fall in the range 9 to 15 over the 33.4 year period, which looks normal.  For example, 328 had 13 hits, 493 had 11, and 473, 12.  Some results that some here will not want to believe are 111:12 hits, 888:13, and 000:14. The results are all public information at the PA Official site.

                Which brings us to some more complicated backtests, those that aren't available seamlessly at the PA site.  For starters, I just wrote a tiny program which scans the 11,563 lines in the file to answer some of the easier questions.  The first question I asked was, What would have happened if each day I had bought a straight ticket on the PA Daily Number (Evening) , and religously chose the WINNING NUMBER OF THE PREVIOUS DAY?  So, here is our first SYSTEM in the Fooled by Randomness thread!  Always buy the previous day's winning number.  (Send out a Toad Alert!)  Smile 

                Hopefully this will be the prelude to testing some of the more sophisticated Systems published here for 3 digit lotteries.

                According to the same probability theory used to calculate the number 12 above, for more or less random selections, the results for yesterday's winners should be the same!  On any given day, in a fair game, the odds of a specific number hitting is 1:1000.  Now, will this apply to a choice like "Yesterday's Winner?"  We will soon see.  But first, I'd like to satisfy my curiosity by staging a Poll here.  I've never done one, so I'm not sure where it will be seen when I initiate it from this Forum/Thread.  This site is pretty cool, so I'm sure it will work out.

                Look for the Poll soon!

                "Theoretically, according to probability as I learned it, a particular 3 digit number is expected to be drawn, on average, once every 1000 draws, provided there is no human or mechanical interference in the selection process."

                I have a program that tracks past result but doesn't include all the past drawings so I could only look at the past 5400 in the PA evening drawing. The mean would be 5 draws so I counted the numbers with less than 5 hits and there were 395. This tells us that about 40% of the numbers didn't average 1 hit every 1000 drawings. And 20 of those numbers only hit once so while probability says each number, including the previous drawing should hit once every 1000 drawings, it doesn't mean they will.

                On the other side of the coin, 6 numbers did repeat so your starter system did do better than once every 1000 drawings in that time frame!


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                  Posted: August 8, 2010, 1:27 pm - IP Logged

                  "Theoretically, according to probability as I learned it, a particular 3 digit number is expected to be drawn, on average, once every 1000 draws, provided there is no human or mechanical interference in the selection process."

                  I have a program that tracks past result but doesn't include all the past drawings so I could only look at the past 5400 in the PA evening drawing. The mean would be 5 draws so I counted the numbers with less than 5 hits and there were 395. This tells us that about 40% of the numbers didn't average 1 hit every 1000 drawings. And 20 of those numbers only hit once so while probability says each number, including the previous drawing should hit once every 1000 drawings, it doesn't mean they will.

                  On the other side of the coin, 6 numbers did repeat so your starter system did do better than once every 1000 drawings in that time frame!

                  And what does this lead you to conclude?

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                    NASHVILLE, TENN
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                    Posted: August 8, 2010, 2:28 pm - IP Logged

                    I have a program that tracks past result but doesn't include all the past drawings so I could only look at the past 5400 in the PA evening drawing. The mean would be 5 draws so I counted the numbers with less than 5 hits and there were 395. This tells us that about 40% of the numbers didn't average 1 hit every 1000 drawings. And 20 of those numbers only hit once so while probability says each number, including the previous drawing should hit once every 1000 drawings, it doesn't mean they will.

                    Because your results are skewed and are opposite what theory says should be, I went to the PA website to find out if the game was ball drop or RNG.  While the information was not straightforward, I am lead to believe that particular game is RNG.   Feel free to correct me if I am mistaken.

                    Being RNG, is anyone surprised by the skewed results?  Should you be?

                    This, as far as I am concerned, is further evidence that states manipulate RNG to their benefit.


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                      Posted: August 8, 2010, 8:39 pm - IP Logged

                      I have a program that tracks past result but doesn't include all the past drawings so I could only look at the past 5400 in the PA evening drawing. The mean would be 5 draws so I counted the numbers with less than 5 hits and there were 395. This tells us that about 40% of the numbers didn't average 1 hit every 1000 drawings. And 20 of those numbers only hit once so while probability says each number, including the previous drawing should hit once every 1000 drawings, it doesn't mean they will.

                      Because your results are skewed and are opposite what theory says should be, I went to the PA website to find out if the game was ball drop or RNG.  While the information was not straightforward, I am lead to believe that particular game is RNG.   Feel free to correct me if I am mistaken.

                      Being RNG, is anyone surprised by the skewed results?  Should you be?

                      This, as far as I am concerned, is further evidence that states manipulate RNG to their benefit.

                      GASMETERGUY,

                      PA "Daily Number" MidDay is computer selected, the Evening draw is Live Ping Pong Balls.  I'm using the (Evening) data.

                      Please double check me on that, but I'm pretty sure that's the way it is.  (I wouldn't have chosen that game if I new it was computerized.)

                      We watch the Daily Number (Evening) here on Local TV at 7PM.

                      Skewed?  What do you mean?  You looked at 5,400 drawings, I'm looking at nearly 12,000. This is why I'm expecting 12 hits, and you should be expecting 5.  I also expect there will be numbers that hit more or less than 12 times(5 in your case.)  Did you count how many hit MORE than 5 times?  If I have time, I will calculate the mean and standard deviation of the "Number of Hits" for each of the 1000 numbers (000-999) in my Aprox. 12,000 draws.  That result plus the distribution of the hits resulting from playing "Yesterday's Winner" should clear this up for you.

                      --Jimmy


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                        Posted: August 8, 2010, 9:46 pm - IP Logged

                        The name of this Thread is the name of one of my favorite books, written by Nicholas Taleb.  If you read it, you will know why I chose This Title, in This Forum.  It's too late for me to elaborate further at this time.  Tomorrow is another day.

                        Harry C. Edwards, a reviewer at Amazon, had this to say:

                        "If the prescriptions for getting rich that are outlined in books such as The Millionaire Next Door and Rich Dad Poor Dad are successful enough to make the books bestsellers, then one must ask, Why aren't there more millionaires? In Fooled by Randomness, Nassim Nicholas Taleb, a professional trader and mathematics professor, examines what randomness means in business and in life and why human beings are so prone to mistake dumb luck for consummate skill. This eccentric and highly personal exploration of the nature of randomness meanders from the court of Croesus and trading rooms in New York and London to Russian roulette, Monte Carlo engines, and the philosophy of Karl Popper. Part of what makes this book so good is Taleb's ability to make seemingly arcane mathematical concepts (at least to this reviewer) entirely relevant in evaluating and understanding everything from the stock market to the success of those millionaires cited in the aforementioned bestsellers. Here's an articulate, wise, and humorous meditation on the nature of success and failure that anyone who wants a little more of the former would do well to consider."

                        --Highly recommended

                        --Jimmy

                        OK All,

                        I'm pressed to go to my other life shortly, so this will be shorter than I planned.  (I'm sure you all know about that problem!)  I will elaborate more later or tomorrow if comments warrant it.  On the Poll, the correct answer was 12.  7 out of 9 people got it right, which is great!  I'd like to know why people picked the 3 and 100 options.  Please don't be reluctant.  Rest assured, you are not alone.  Books have been written about this.  Take the title of this Thread, for example!  Smile

                        You will be justified if you complain that in the Poll Question I did NOT specify the number of hits EXPECTED, which is what 12 is.  In reality, with this specific Lottery data, the actual result was 14!  So Shoot me!  Unhappy  I think everyone will still see the point here.

                        I really have to leave now, so let me ask another Poll qustion, without a formal Poll:  What do you think the results will look like if we change the "System" to be: Every day, play the winner from two(2) days ago (instead of one)?   (This is a hypothesis, not a command!)

                        I also promised someone in another thread that I would calculate the Mean & Standard Deviation of the "Number of Hits" for each of the 1000 numbers, over the 12,000(approximate) Draws.  That will be in the next post along with the results for the above question.

                        Please alert me to anything that would indicate a problem with my data, or programs.  You know the old saying, "Garbage in, garbage out!"

                        --Jimmy

                        ---------------

                        PA Lottery Daily Number (Evening) [3 digits, 000-999]

                        1977/03/01 thru 2010/08/08

                        Playing Yesterday's Winning Number

                        1980/01/09: 182
                        1980/01/10: 182  HITS== 1
                         
                        1989/01/18: 430
                        1989/01/19: 430  HITS== 2
                         
                        1989/07/06: 144
                        1989/07/07: 144  HITS== 3
                         
                        1993/09/29: 400
                        1993/09/30: 400  HITS== 4
                         
                        1994/02/13: 956
                        1994/02/14: 956  HITS== 5
                         
                        1998/08/16: 697
                        1998/08/17: 697  HITS== 6
                         
                        1998/10/14: 792
                        1998/10/15: 792  HITS== 7
                         
                        2001/04/14: 505
                        2001/04/15: 505  HITS== 8
                         
                        2002/05/17: 415
                        2002/05/18: 415  HITS== 9
                         
                        2005/04/07: 878
                        2005/04/08: 878  HITS== 10
                         
                        2006/12/19: 620
                        2006/12/20: 620  HITS== 11
                         
                        2007/09/01: 798
                        2007/09/02: 798  HITS== 12
                         
                        2007/09/17: 882
                        2007/09/18: 882  HITS== 13
                         
                        2009/09/12: 407
                        2009/09/13: 407  HITS== 14

                        Later!


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                          Posted: August 9, 2010, 2:56 am - IP Logged

                          The Quest for a Blockbuster System

                          In a couple of my earlier posts, I mentioned that I had 12,200 Draw results for the Pennsylvania Daily Number game (Evening).  This was incorrect.  I hadn't finished grabbing all 33.4 years from the PA website, so I multipled 33.4 by 365 and rounded to 12,200.  In fact, PA did not draw Daily Numbers on Sundays for nearly 2 years.  So, now that I'm done, what I have are 11,563 rows from March 1, 1977 to today, August 3, 2010.

                          Theoretically, according to probability as I learned it, a particular 3 digit number is expected to be drawn, on average, once every 1000 draws, provided there is no human or mechanical interference in the selection process.  This theory can be tested easily at the PA website by merely entering the number of interest in the Daily Number (Evening) "Has my number ever hit before" boxes, and clicking Search.  Now, since there have been approximately 12,000 draws, we should expect each number to be drawn around 12 times, give or take.  The give or take is measured in standard deviations, which I have not yet calculated.  However, more often than not, I observe the number of hits to fall in the range 9 to 15 over the 33.4 year period, which looks normal.  For example, 328 had 13 hits, 493 had 11, and 473, 12.  Some results that some here will not want to believe are 111:12 hits, 888:13, and 000:14. The results are all public information at the PA Official site.

                          Which brings us to some more complicated backtests, those that aren't available seamlessly at the PA site.  For starters, I just wrote a tiny program which scans the 11,563 lines in the file to answer some of the easier questions.  The first question I asked was, What would have happened if each day I had bought a straight ticket on the PA Daily Number (Evening) , and religously chose the WINNING NUMBER OF THE PREVIOUS DAY?  So, here is our first SYSTEM in the Fooled by Randomness thread!  Always buy the previous day's winning number.  (Send out a Toad Alert!)  Smile 

                          Hopefully this will be the prelude to testing some of the more sophisticated Systems published here for 3 digit lotteries.

                          According to the same probability theory used to calculate the number 12 above, for more or less random selections, the results for yesterday's winners should be the same!  On any given day, in a fair game, the odds of a specific number hitting is 1:1000.  Now, will this apply to a choice like "Yesterday's Winner?"  We will soon see.  But first, I'd like to satisfy my curiosity by staging a Poll here.  I've never done one, so I'm not sure where it will be seen when I initiate it from this Forum/Thread.  This site is pretty cool, so I'm sure it will work out.

                          Look for the Poll soon!

                          On with the Quest for the Blockbuster System!

                           

                          First, here are the results of betting on the Two(2) day old Winner:

                          (This is PA Daily Number (Evening) [000-999] 3/01/1977 thru 08/08/2010)

                          (Remember, this lottery did not have daily draws in all years - Note the 1st entry.)

                          1980/06/21: 772
                          1980/06/23: 124
                          1980/06/24: 772  HITS== 1
                           
                          1982/02/09: 695
                          1982/02/10: 268
                          1982/02/11: 695  HITS== 2
                           
                          1985/02/20: 918
                          1985/02/21: 682
                          1985/02/22: 918  HITS== 3
                           
                          1986/12/16: 710
                          1986/12/17: 999
                          1986/12/18: 710  HITS== 4
                           
                          1988/03/16: 823
                          1988/03/17: 456
                          1988/03/18: 823  HITS== 5
                           
                          1988/05/27: 870
                          1988/05/28: 617
                          1988/05/30: 870  HITS== 6
                           
                          1989/04/10: 396
                          1989/04/11: 597
                          1989/04/12: 396  HITS== 7
                           
                          1993/01/09: 854
                          1993/01/10: 816
                          1993/01/11: 854  HITS== 8
                           
                          1993/06/18: 119
                          1993/06/19: 844
                          1993/06/20: 119  HITS== 9
                           
                          1993/11/30: 836
                          1993/12/01: 546
                          1993/12/02: 836  HITS== 10
                           
                          1995/05/10: 092
                          1995/05/11: 587
                          1995/05/12: 092  HITS== 11
                           
                          1996/10/25: 907
                          1996/10/26: 385
                          1996/10/27: 907  HITS== 12
                           
                          2002/08/24: 867
                          2002/08/25: 433
                          2002/08/26: 867  HITS== 13
                           
                          2003/03/23: 354
                          2003/03/24: 585
                          2003/03/25: 354  HITS== 14
                           
                          2007/01/09: 658
                          2007/01/10: 198
                          2007/01/11: 658  HITS== 15
                           
                          2008/04/20: 694
                          2008/04/21: 798
                          2008/04/22: 694  HITS== 16

                          Anyone want to see the results for betting on the THREE(3) day old Winner?

                          ----------------------------------------

                          And as promised, here are the Summary Stats of the Raw data.

                          These summary statistics are measures made using the raw

                          PA Daily Number (Evening) [3 Digit 000-999] drawing results.

                          March 1, 1977 thru August 8,2010

                          Parameter Measured:  Number of Occurences of each of the numbers [000---999]

                          Software:                     MS-Basic & Analyze-It (Excel Plug-in)

                          Total Draws:                11,567

                                                                                 95% Confidence Interval

                          Mean                      11.6                  11.4 -- 11.8

                          Median                   11                     11 -- 12 

                          Std. Dev                  3.3                   3.1 -- 3.4

                          -----------------------------------------------------------

                          Range                     20

                          Percentiles

                          0th                          3                   (minimum)

                          25th                        9                   (1st quartile)

                          50th                       11                  (median)

                          75th                       14                  (3rd quartile)

                          100th                     23                  (maximum)

                          ------------------------------------------------------------

                          The Bar Charts and Normality Fit Graph are beautiful.

                          This is very Normal Data!

                          With 11,567 Draws, I'd say a mean of 11.6 is PRETTY DARN CLOSE!

                          ------------------------------------------------------------

                          Cross Check Interpretation of the Normal Distribution:

                          A Normal distribution with a mean of 11.6 and a Standard Deviation of 3.3

                          should have the following characteristics. (From any stat textbook)

                          (Rounding to Integers)

                          68% of the (# of Hits) fall in the range 8 to 15,

                          95% of them fall between 5 and 18, and

                          99.7% of them fall between 2 and 22.

                          (Comparing)

                          The min and max actually observed were 3 and 23.

                          This closely agrees with the Analyze-It Percentiles above.

                          Only one number had 23 Hits ===> 308.   It was one step outside the

                          3rd Std. Dev and I thought somebody might be curious about it.

                          -------------------------------------------------------------

                          These are the summary numbers based on the ACTUAL observed

                          draws of this PA Lottery.  What I am calculating for my next post is this

                          same set of statistics applied to 11,567 Pseudo Random Draws!   Smile 

                          I know many of you don't "believe in" or "like" RNGs, but wouldn't you be

                          interested in how the stats derived from actual Draws compare to

                          Pseudo Random Draws?  Actually, I already did it.  It simply required a

                          replacement of a read of a line from the actual Draw file with a call to my RNG.

                          While you're digesting this, I'll work on the next post, which will be a side-by-side

                          comparison of the above stats from each of the 2 data sources, ACTUAL and RANDOM.

                           

                          Can you guess what the comparison will look like?

                           

                          By the way, I invite anyone who is skeptical of this reporting to

                          duplicate my effort, either with the same PA data I'm using, or any [000-999]

                          data you have from a state you're familiar with.  I wish you WOULD use my

                          same data, so there would be no controversy over the results.  If you use the

                          PA Daily Number (Evening) 1977-Date, you should get the results above.

                          All it takes is 33 Copy & Pastes of the Previous Results page at the PA Official

                          site, which is served up by year.

                          Note:

                          For anyone who got excited over the 23 hits of the number 308 above,

                          consider this.  If you bought a $1 ticket on 308 every day for the 30+ years,

                          you would have spent $11,567.00.  Your 23 hits would have netted you

                          $11,500(In PA 23*500), for a NET LOSS of $67.00.

                           

                          --Jimmy

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                            Kentucky
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                            Posted: August 9, 2010, 12:14 pm - IP Logged

                            And what does this lead you to conclude?

                            Had I played the previous number for $1 in 5400 drawings and hit 6 times, I would have lost $2400 because the lotteries have a statistical payoff edge over any pick-3 bet. There probably are money management methods where betting the same number could show a slight profit, but realistically it's better to find something with a higher statistical probability than slightly better than average.

                            Basically you found something in the past that statistically exceeded the average for that time period without any guarantee it will continue into the future.

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                              Kentucky
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                              Posted: August 9, 2010, 1:52 pm - IP Logged

                              I have a program that tracks past result but doesn't include all the past drawings so I could only look at the past 5400 in the PA evening drawing. The mean would be 5 draws so I counted the numbers with less than 5 hits and there were 395. This tells us that about 40% of the numbers didn't average 1 hit every 1000 drawings. And 20 of those numbers only hit once so while probability says each number, including the previous drawing should hit once every 1000 drawings, it doesn't mean they will.

                              Because your results are skewed and are opposite what theory says should be, I went to the PA website to find out if the game was ball drop or RNG.  While the information was not straightforward, I am lead to believe that particular game is RNG.   Feel free to correct me if I am mistaken.

                              Being RNG, is anyone surprised by the skewed results?  Should you be?

                              This, as far as I am concerned, is further evidence that states manipulate RNG to their benefit.

                              The PA evening pick-3 drawing is and has always been done by live ball drawings.

                              There is much more to probability than assuming each number should be drawn one time every 1000 drawings and the results show this. Looking at the last 1000 PA evening drawings, the digit 9 appeared 81 times in the first position meaning it was impossible for for at least 19 numbers starting with the digit 9 to be drawn in those 1000 drawings. The digit 2 was drawn 91 times and 6 was drawn 92 giving another 17 numbers. In the second digit position there were 5 digits failing to be drawn 100 times so add at least another 34 numbers. The third position had at least another 33 numbers for a total of at least 103 numbers (10%) that had no chance of being drawn.

                              Simply put, any digit in any position must be drawn at least 100 times before all 100 numbers using that digit can be drawn. It's cause and affect because if one digit in any position is drawn more than average at least one other digit will be drawn less than average meaning it's impossible for at least one 3 digit number to be drawn.

                              Comparing PA's live ball drawing results to TN RNG drawing, PA had at least 103 impossible numbers and TN had 111. To get the best comparison we would need to compare all the live ball drawing with all the RNG drawings, but for now comparing two is should be sufficient because of the time it would take.

                                 
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