Originally Posted by
ssanders82
Awesome, thank you. I may be missing something from that post, but can you explain how you calculated the std dev of 1000 coin flips as 15.81 heads? Without going into a bet-by-bet variance analysis I'm trying to calculate % chances that I have a winning system after n bets and x wins.
After n Bernoulli trials (i.e., trials of a binary outcome event -- in this case heads/tails) with success rate p, variance is given by:
σ2 = n * p * (1-p)
and hence standard deviation:
σ = √n * p * (1-p)
So in the case of 1,000 trials of a coin flip:
σ = √1,000 * 0.5 * (1-0.5) ≈ 15.81
But if all your bets have roughly the same breakeven probability anyway, then you're probably best served determining the Type I error of the hypothesis that you're no better than a 0 EV handicapper directly from the binomial distribution.
So if you've placed N bets at breakeven probability p and have won W of them, the probability that you'd see these results or better were you in fact only a 0 EV bettor would be given in Excel by:
=1-BINOMDIST(W-1,N,p,1)
So let's say you've placed 200 bets at -110 and have won 109 of them, the probability that these results would be seen by chance alone had you in fact no advantage would be given in Excel by:
=1-BINOMDIST(109-1,200,110/210,1) ≈ 29.87%
meaning that you could not reject the null hypothesis at α = 0.05 that you were no better than 0 EV.
Performing the same analysis using the Normal distribution:
σ = √200 * 110/210 * (1-110/210) ≈ 7.06
Zone tailed ≈ (109-200*110/210)/7.06 ≈ 0.60
p(Z ≥ 0.60) ≈ 27.42%
which due to the flat tails of the normal distribution implies slightly higher significance than that of reality.