Re: variance of shoe

From Stanford Wong's BJ21

Posted by MathProf on 13 Dec 1997, at 6:04 a.m., in response to variance of shoe, posted by Vic Bass on 13 Dec 1997, at 3:13 a.m.

If you are asking what the Vraince in Counter winnings are, this depends on the SPread. You can find data like this in BJ Attack.

However I thought you were asking what is the variance of the True Count in a Shoe. YOu can compute this directly. I'll illustrate this with Hi-Lo.

First for Hi-LO, the "Variance" of a card is 10/13. This should more properly called the Sum of Squares, or Expecetd Square. Because this is a balanced count, the SSW and Var will be the same; however for an Unbalanced Count, they will not be. YOu should then still use SS, which is easier to compute. I will call this SS1, for Sum of Squares of 1 card.

Now suppose we had a 1-deck remainder, which had 52 cards, and suppose these had been selected from an infinite deck (sampling with replacement). Then the Variance of the Running Count there will be (52)*(10)/(13). More generally it will be n*SS1.

However we are not sampling with replacement. If we are pulling from a finite deck, then we have to make an adjustment. The formula is 


  Variance   =  [ (N-n) / (N-1) ]  *  IndepVar

             =  [ (N-n) / (N-1) ] n * SS1  

  where   N is Total cards in Shoe.

          n is the size of the remainder.

BTW, this formula is symmetirc with respect to n and N-n. That is, if we are sampling 2 decks from 6, we will get the Variance (of Running Count) as if we sampled 4 decks from 6, as we should.

Let us do an example. 6 decks dealt to 2. HiLo Count. We are sampling 104 cards, if it were done with replacement, the Variance would be 104*10/13=80. However, we are sampling with replacement. Our correction factor is (312-108)/(312-1) because 312 card is 6 decks. Note that this correction factor is approximately 4/6. 4 is the Number of Decks that have been dealt. I always use this approximation when doing these calculations ... Anyways this give us approximately 53.3 Variance. (4/6)*(10/13). Please Note this is the Variance of the Running Count, the SD is the square root which is 7.3. Now this is the SD of the RC. Since we are at 2 decks, the SD of True COunt is 7.3/2 which is about 3.7. (Note that you cannot divide the RC-Var by the Number of Decks; you must divide the RC-SD by the Number of decks).

This is the SD at 1 point. To get it thru the entire deck, you have to add them up at the various points along the way. YOu could probably set up an INtegral toa pproxiamte this assumming that all penetratiion elvels are equally likely)

Having said all that, let me note that this distribution is not Normal. If we look at the distribution at, say the 2 deck level as in the example, the curve is "fatter" than a Normal distribution. (more Kurtosis). So Counts that are over 1 SD away will occurr more often than with the Stdrd Normal.

Now having said all that, let me note that if you want to know the TC distribution for Real Shoes, you can see this in the charts of BJ Attack. DOn lists the distribution for positive TC's, and hjsa the negatives all together. However the distribution is symmetric, so the negatives have the same distribution as the positives.
One final disclaimer: Everything I said will work for any balanced Count. Just use the appropriate SS1. I think it will also work for unbalanced counts, but I am not quite sure, I have to think about that a little more. I haven't done much with Unabalnced counts. I suffer from Mathematicans "balanced Count myopia".

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