Variance of True Counts

This may not be exactly what you want, but I think it will go a long way toward answering your question.

The Variance (and standard deviation) of the True Count at various levels is given by a relatively simple formula. If N is the number of cards in the pack, and R is the number remaining cards (so that N-R) is the number which has been dealt, the the variance of the running count is given by

where s is a constant that depends on the count system you are using. For HiLo, s is 10/13.

You can get the SD or the RC by taking the square root of the formula given above.

Note that our formula is symmetric between R and N-R. That is, the distribution of running counts after 1 deck ash been dealt is the same as it distribution when only 1 deck remains. (Do you see why ? ) This is obviously NOT the case for true counts.

We get the Var of the TC by dividing by R^2, and get

I put all the constant together out front. The function (N-R)/R increases as R gets small.

By the way, I think these are the per card True Counts. If you want per deck TCs (which is what most of us use), you need to put in a factor of 52.

Thanks.

"(Do you see why ? )", "This is obviously NOT the case for true counts.", and " The function (N-R)/R increases as R gets small." each indicate a great teacher has written.

My knowledge of anything related to statistics and game theory is always doubtful. Here is what I understand:

It seems that the reason why they should be the same is that they are on opposites sides/ends of the tally of possible results at equal points from half. Or to put it even more crudely, one is what is found if one breaks off a piece of what started as zero and the other is the starting chunk without the broken off piece. Right?

But on examining the formula I find that at say one deck used and one deck left: The non-constant multiplicand and multiplier are the same but have switched positions; by some axiom whose name I forget the product will be the same. However it seem that the divisor would be different and thus the result should be different.

What have I missed?

I presume that S is the same as s.

Slot
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Var (RC) = s * R*(N-R) / (N-1)

where s is a constant that depends on the count system you are using. For HiLo, s is 10/13.

CUT

Note that our formula is symmetric between R and N-R. That is, the distribution of running counts after 1 deck ash been dealt is the same as it distribution when only 1 deck remains. (Do you see why ? ) This is obviously NOT the case for true counts.

We get the Var of the TC by dividing by R^2, and get

Var(TC) = S/(N-1) * (N-R)/R

N is just the total number of cards in the shoe, so (N-1) doesn't change.

Say you're dealing with a 6 deck shoe, and you want to compare the variance under hi-lo of 1 deck used, and 1 deck remaining. Then for 1 deck used we have s = 10/13, N = 312, and for one deck used R = 52. For 1 deck remaining R = 312 - 52 = 260.

Var (RC) = s * R*(N-R) / (N-1)
Var(RC at 1d) = 10/13 * 52*(312-52) / (312-1) = 33.4405...
Var(RC at 5d) = 10/13 * (260)*(312-260) / (312-1) = 33.4405...

Let's see, if R2 = N-R1, then R2*(N-R2) = (N-R1)*(N-(N-R1)) = (N-R1)*R1 = R1*(N-R1). Commutative property of multiplication (or something like that ;-}).

ETF

Of course. But now I wonder where my mind was when I looked at this before. Thanks.

Slot