If you want to know with a 95% confidence that your win rate lies between x- m.o.e and x + m.o.e , where x is your sample mean and m.o.e is your margin of error ,then use the following approximation formula :
Total number of games = (1/M.O.E)^2
So if your sample mean is 55 % and you're interested in the total number of games to determine if you're a winning player , then we would use a m.o.e of 2.5% since 52.5% is the break even point . This means that you would need to play (1/0.025)^2 = 1600 games to be 95 % certain .
Here is another example :
Suppose you've played 500 games and won x% of games but you want to be 95% confident that your win rate lies between x-m.o.e, x+m.o.e , then we would solve the equation .
Total number of games = (1/m.o.e)^2 500=(1/m.o.e)^2 m.o.e=sqrt(1/500) m.o.e = 4.47 %
So with 95% confidence , we know our confidence interval to be x - 4.47 , x + 4.47 . So if x =60% , then our interval is 55.53,64.47 .