The average profit per game of the player in dependece on the number of games which were played. 

Game[n_]:=Table[Module[{s=1},While[Random[Integer]==1,s=2*s];s],{k,n}];
AverageProfit[n_]:=N[Apply[Plus,Game[n]]/n];
ListPlot[Table[AverageProfit[k],{k,300}]]