Monday, April 11, 2016

The Galton Board (Quincunx)

   The Galton Board (quincunx) was invented by Sir Francis Galton (1822–1911) as a mechanical device to demonstrate the 'normal distribution' in statistics.
   "He had noticed that a normal curve is reproduced by lead shot falling vertically through a harrow of pins…" For more information visit the following link.
   The Galton Board is nothing more than a gravity-powered random walk on the integer number line where the walker, starting at zero, flips a coin to move one unit to the right if a head or one unit to the left if a tail. The probability of flipping a head is 1/2 and the probability of flipping a tail is also 1/2 
   In a Galton Board, the coin flipping is replaced with a marble striking a peg and then going left or right with equal probabilities.
   There is a neat programming shortcut used to determine the bin number (1 – 10) that each ball falls into  after striking the ninth peg. One just has to count the total number of 'lefts' (or 'rights') at the end of the ninth bounce. The order of lefts or (rights) is not important. At the bottom of each numbered bin. Click on 'Show List' to view the totals for each bin.
                                     Bin Number
                1     2     3     4     5     6     7     8     9     10
                Random Walker Position on Number Line
               -9   -7    -5     -3  -1   +1   +3   +5   +7   +9

                Theoretical Values for Each Bin (9 flips)
                                       Bin Number
               1      2      3      4      5      6      7      8      9     10
               1      9     36   84   126  126    84    36     9      1

                 One Run of the Project set to 512 Marbles 
                                        Bin Number
               1     2      3     4     5       6      7     8     9     10
                0    4    34    97  127   129   73    35    9      4

      This project can be viewed and downloaded at this link:
    This project complements my other Random Walk projects:
   The Huckster's Game
   Feynman’s Random Walk
   Random Walk with Barriers
   Jean Perrin's Random Walk Experiment
   Random Walk on the Integer Number Line Probabilities