pe0 = PDF[BinomialDistribution[99, 1/2], nE]

Solve[pe == (1 - pe)/2, pe]

{{pe -> 1/3}}

start = RandomChoice[Range[6], 99];
BarChart[Apply[Labeled, Reverse[Sort@Tally[start], 2], {1}]]

f = Block[
   {rndIndx = RandomInteger[{1, 99}], new},
   new = If[EvenQ@Part[#, rndIndx], 1, RandomInteger[{1, 6}]];
   ReplacePart[#, rndIndx -> new]
   ] &

reslist = NestList[f, start, 100000];
evol = Transpose[(#/Total[#]) &[Part[Sort@Tally[EvenQ[#]], All, 2]] & /@ reslist];
ListLogLinearPlot[evol, Joined -> True, Epilog -> {Line[{{0, 1/3}, {100000, 1/3}}], Line[{{0, 2/3}, {100000, 2/3}}]},
 PlotRange -> {{1, 100000}, {0, 1}},
 Frame -> True
 ]

Sort@Tally[Last@reslist]

{{1, 37}, {2, 9}, {3, 10}, {4, 17}, {5, 15}, {6, 11}}

BarChart[Apply[Labeled, Reverse[%, 2], {1}]]

N[Mean /@ evol]

{0.666522, 0.333478}

N[StandardDeviation /@ evol]

{0.048476, 0.048476}

Clear[a, n, m, even];

even[1] = n/4;

even[m_] := even[m] = (n - even[m - 1])/2;

Take[Table[even[m], {m, 1, 100}], -3] // N

Clear[even]

even[m_] =
 a[m] /. RSolve[{a[m] == (n - a[m - 1])/2, a[1] == n/4}, a[m], m][[1]] //
  ExpandAll

even[m] == n/3 (1 + (1/2)*(-1/2)^m) //
 Simplify[#, {Element[m, Integers], m > 0}] &

Limit[n/3 (1 + (1/2)*(-1/2)^m), m -> Infinity]