Solving for standard deviation

Hi,

I'm writing a simulator for solving the standard deviation of wagering X dollars on pontoon (a blackjack variant with a lower house edge than any form of BJ regularly offered). However, I've tripped over a conceptual error that I'm unsure how to resolve.

The number of bets placed on a hand is not constant, because you can double or split. Since the pdf will end up being normal, I supposed that I could use the formula B*S*sqrt(WR/(B*M)), where:
B is the bet size, S is the s.d. of a single hand of pontoon, WR is the amount we are wagering, and M is the expected number of bets placed on a single hand of pontoon.

Unfortunately, there is a covariance between the number of hands played and the average bet size: the more we double and split, the less hands we will play. How do I resolve this problem? (Thanks to the person who pointed this out to me.)

I also wonder, as a result of this covariance, if this problem even has an exact solution at all. I've always used WoO's value of 1.16 s.d. on blackjack to find the standard deviation of 100 bets by doing 1.16*sqrt(bets). Does this 1.16 account for the covariance, or is it just an inaccuracy that we have to live with?

Thanks for any and all help!

[alThor]

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I've always used WoO's value of 1.16 s.d. on blackjack to find the standard deviation of 100 bets by doing 1.16*sqrt(bets). Does this 1.16 account for the covariance, or is it just an inaccuracy that we have to live with?

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That is the SD for one hand. Therefore it cannot have anything to do with covariance.
http://wizardofodds.com/blackjack/bjapx4.html

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I'm writing a simulator for solving the standard deviation of wagering X dollars on pontoon (a blackjack variant with a lower house edge than any form of BJ regularly offered).

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If you're simulating, I'm not really sure what the problem could be. Keep track of the total dollars won/lost on each round of play. Also keep a running total of the square of that number. At the end, variance is the sum of squares(*) minus the mean squared, and you're done. Maybe you didn't mean to say simulation?

alThor

(*) I'm intentionally ignoring the "n/n-1" factor.

[Siegmund]

I think it's confusion over what is meant by "one hand." If you split a pair, you now choose what action to take on each half and have two outcomes you add together. If you count a split as 'two hands' you have a situation where the number of hands you play per initial bet depends on your strategy, and you might make additional bets before you are done acting on your hand.

But I think the common usage is that when we calculate the EV and SD per initial bet, we count the complete series of events that follows including splits and doubles and insurance as "one hand" despite the fact that more chips might be committed beyond the initial bet.

I AM running a simulation of one million hands, and I know how to calculate the variance; and I'm counting splits as one hand. My problem is that I'm solving for the standard deviation of one HAND, but then trying to use that to find the standard deviation of a hundred WAGERS, which could number anywhere from one to, say, 1.5 per hand. Therefore, I can't use the usual s.d.*sqrt(hands), because the number of hands will be a variable. I also can't do s.d.*sqrt(hands/avg # bets per hand), because there is a covariance between the number of hands and the average number of bets placed per hand.

To put it another way, I don't play exactly H/avg hands, but rather H/avg +- something, which gives an additional term to the standard deviation. If I finish in less than H/avg hands, that means I did better because I doubled and split more often. So there's a negative correlation, and I need to account for that covariance to get the exact standard deviation of placing N bets.

[alThor]

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My problem is that I'm solving for the standard deviation of one HAND, but then trying to use that to find the standard deviation of a hundred WAGERS

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Yes, and that's what cannot be done (if I understand your use of those terms). So can't you incorporate all of this extra uncertainty (# hands, wager size, etc.) into your simulation engine? If so, you can estimate all kinds of things regarding the risk per ROUND of betting.

alThor

[Izverg04]

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If so, you can estimate all kinds of things regarding the risk per ROUND of betting.

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He doesn't want to calculate risk per ROUND of betting, he wants to calculate risk per BET.

He's got a dataset with 2 columns. Column1 has NBETS (bets put in on a round), Column2 has RESULT. He wants to use this dataset to calculate the variance on 1 bet, not 1 round.

A simplistic method to do this is to calculate Var(Column2)/Ave(Column1). This is wrong though (although not by much to be important in practice). To get the right answer, you also need to incorporate Var(Column1) and Cov(Column1,Column2) which is non-zero.

I'd have to spend a bit of time to give the OP the right answer, but I hope the problem is now made clear.