Count preponderance of 1s in batches of 165, with all possibly initial shifts. Sole exception: shift=23. See RNGwalkwithshift.out.txt

RNGwalkwithshift.tex

\newcount\cnta
\newcount\cntb
\newcount\cntc
\newcount\cntd

\newcount\maxrand
\maxrand "10000000

% \the\maxrand

\input xinttools.sty

\def\testwalk#1#2#3{%
   % #1 = seed (O or 1)
   % #2 = number of repetitions
   % #3 = length of batches; must be an odd integer
   \def\Dzero{0}%
   \def\Done{0}%
   \pdfsetrandomseed #1
   % do \foo times a call to \pdfuniformdeviate\maxrand
   \cnta 0
   \xintloop
     \ifnum\cnta<\foo\space
     \edef\x{\pdfuniformdeviate \maxrand}%
     \advance\cnta 1
   \repeat
   % now do #2 times:
   % examine if more 1s than 0s in next #3 \pdfuniformdeviate \maxrand
   \cnta 0
   \xintloop {%
     \def\S{0}%
     \cntb 0
     \xintloop
       \ifodd\pdfuniformdeviate\maxrand\relax
         \edef\S{\the\numexpr\S+1}%
       \else
         \edef\S{\the\numexpr\S-1}%
       \fi
     \advance\cntb 1
     \ifnum\cntb < #3
     \repeat
   }\ifnum\S>0
       \edef\Done{\the\numexpr\Done+1}%
    \else
       \edef\Dzero{\the\numexpr\Dzero+1}%
    \fi
    \advance\cnta 1
    \ifnum\cnta < #2
    \repeat

    \immediate\write\out{(shift=\foo, seed=#1) 0:\Dzero, 1:\Done}%
}

\newwrite\out
\immediate\openout\out=\jobname.out

% Try with all 55 possible starting point
% (the special case "54" makes theoretical analysis a bit easier,
%  when batches have a length multiple of 55)

% We only study runs of 165 draws here.

\immediate\write\out{+++++++++++++++++++++++}
\immediate\write\out{10000 runs of 165 draws}
\immediate\write\out{+++++++++++++++++++++++}
\immediate\write\out{}

\cntc 1

\xintloop
\edef\foo{\the\cntc}%
\testwalk {0}{10000}{165}%
    
\testwalk {1}{10000}{165}%

\immediate\write\out{}%

\ifnum\cntc<55
\advance\cntc 1
\repeat

\immediate\write\out{}
\immediate\write\out{++++++++++++++++++++++}
\immediate\write\out{30000 runs of 55 draws}
\immediate\write\out{++++++++++++++++++++++}
\immediate\write\out{}

\cntc 1

\xintloop
\edef\foo{\the\cntc}%

\testwalk {0}{30000}{55}%
    
\testwalk {1}{30000}{55}%

\immediate\write\out{}%

\ifnum\cntc<55
\advance\cntc 1
\repeat


\immediate\closeout\out

\bye