Quote from: Rannasha on September 22, 2013, 04:24:08 PM

Quote from: usagi on September 22, 2013, 03:43:15 PM

You are correct. But the problem is much much deeper than this. Let me begin by asking the simple question; since the KC maximizes profit over number of bets, and since variance decreases with bet size, how many max bets would we need to make in order to decrease variance to get, say, 0.9% < profit < 1.1% assuming all bets were max bets? Going with a set max bet size, say 500 BTC, guarantees we will find a sample size more than sufficient to limit profit in this way. Let's further simplify by going after the RNG and not the house edge.

Variance definitely does not decrease with bet size (assuming total amount wagered is kept constant). The standard deviation for a binomial process is calculated by:
s = sqrt( n * p * (1 - p) )
(s = standard deviation, n = number of samples, p = probability)
If every bet is the same (size b), we can obtain the standard deviation for the total profit (S):
S = b sqrt( n * p * (1 - p) )
If we increase bet-size (b), we decrease n (number of bets) proportionally. But since S is proportional to b and proportional to sqrt(n), we see that with increased bet-size, the standard deviation of the expected total profit goes up.

Which is exactly what he said. Lol.
If you increase bet size, you increase variance.
If you decrease bet size, you decrease variance.

=> Variance is decreasing with bet size.

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