Sonny
Well-Known Member
Okay, so maybe the title needs some work. If I were Arnold Snyder, perhaps it would be titled "Algebraic Approximations of Normal Distributions in Single-Pass Riffle Co-Minglings and Disbursements of Cards in the game of 21."
Maybe my title isn't so bad.
I've read several "Shuffle Tracking For Dummies" type articles, but have always found the techniques to be too difficult to apply in a casino environment. I decided to start looking for a way to simplify and optimize the current methods of shuffle tracking in order to facilitate their use in casino play. Why should the "dummies" have all the fun? Us imbeciles wanna win besides! The following will summarize my findings on shuffle tracking - specifically what Mason Malmuth refers to as "card domination", however I have mostly heard the term "cut-off tracking" used to describe it. This type of tracking is only effective against a single pass shuffle, but variations exist that can be implemented against various shuffles. Hopefully the approximations in this article will help to simplify these cases as well.
Summary of Cut-off Tracking
Cut-off tracking consists of retaining the count at the end of the shoe (after the final round has been played and the cards are about to be shuffled). Assuming a balanced count is employed, the remaining unseen cards (hereafter referred to as the "cut-off slug") must therefore have a value that is equal to the running count but with the sign reversed. For example, if the running count is +7 at the end of the shoe, then the cut-off slug contains cards that will sum to -7 in order to assure a zero final count. The next step in tracking the shuffle is calculating the "average count density" of the used cards in the discard tray (hereafter "discards"). In the above example, assuming a six-deck with five dealt game, you would figure that the discarded 5 decks with a count of +7 would average 7 / 5 = +1.4 per deck. If we then shuffle our cut-off slug with one of the discard decks, we would estimate a count of -7 + 1.4 = -5.6 for the new two-deck shuffled slug. We could then cut these cards to the bottom of the shoe, adjust our starting running count, and play with a significant advantage in a four-deck game. We are essentially using the cut card to "short the deck" of cards we don't want. Similarly, this method can also be used to cut good cards do the top of the shoe.
As we can see, this can be an incredibly powerful tool to use in actual casino play. Unfortunately, the computations can be a bit too clumsy for some of us to have ready when the cut card lands in front of us. Hence the need for a system that can be employed by the average imbecile.
The Approximation Formula
So isn't there an easier way to get from point A to point B? Happily, yes! Let's take a look at the formula we have already:
Shuffled slug = cut-off slug + average count density (per slug)
Average count density = discards / (number of slugs in shoe - 1)
To break this formula down, we will see that the shuffled slug (the value of the cut-off part that we are tracking AFTER the shuffle) equals the original value of the cut-off slug plus the value of the estimated average count of each slug from the discards. The average count is found by dividing the known count (the running count before the shuffle) by the number of slugs it is comprised of (number of slugs - 1). We subtract 1 because we don't want to include the cut-off slug in our division because it has it's own value already.
This is the standard formula which most of you have probably wrestled with while the dealer is shuffling and stacking away. Although it is very straightforward, the division to find the average count density can be difficult when awkward numbers are used. How many of you would have come up with +1.4 in the above example? After a few hours of casino practice, I decided that I couldn't get it. I was having problems with switching the signs as well. I would get confused with the "negative slugs are GOOD now" concept and was afraid that I would cut a bad slug to the front by accident. So I did what anyone with the mentality of a thirteen-year-old boy would do: I whined about it being "too hard" and gave up.
A few months later I sat down with Excel and used the above formula to make a spreadsheet showing different running counts for the cut-off slug and their final outcomes. I thought that having the formula with various solutions in front of me would help me to understand the concepts and perhaps memorize some of the tricky division problems. I figured that memorizing +1.4 is easier than finding 7 / 5. However, after staring at the numbers for a while, something occurred to me. Why am I going through all of this trouble? Why am I swapping signs, subtracting slugs, and dividing "average count densities"? If I have to figure out how many 1.5 deck slugs are in a six-deck game I'll scream! Yes, I know the answer is 4 and it's easy to remember - but when you're starting out and the dealers are using different penetration levels, it can become maddening. That's when I saw the shortcut.
Maybe my title isn't so bad.
I've read several "Shuffle Tracking For Dummies" type articles, but have always found the techniques to be too difficult to apply in a casino environment. I decided to start looking for a way to simplify and optimize the current methods of shuffle tracking in order to facilitate their use in casino play. Why should the "dummies" have all the fun? Us imbeciles wanna win besides! The following will summarize my findings on shuffle tracking - specifically what Mason Malmuth refers to as "card domination", however I have mostly heard the term "cut-off tracking" used to describe it. This type of tracking is only effective against a single pass shuffle, but variations exist that can be implemented against various shuffles. Hopefully the approximations in this article will help to simplify these cases as well.
Summary of Cut-off Tracking
Cut-off tracking consists of retaining the count at the end of the shoe (after the final round has been played and the cards are about to be shuffled). Assuming a balanced count is employed, the remaining unseen cards (hereafter referred to as the "cut-off slug") must therefore have a value that is equal to the running count but with the sign reversed. For example, if the running count is +7 at the end of the shoe, then the cut-off slug contains cards that will sum to -7 in order to assure a zero final count. The next step in tracking the shuffle is calculating the "average count density" of the used cards in the discard tray (hereafter "discards"). In the above example, assuming a six-deck with five dealt game, you would figure that the discarded 5 decks with a count of +7 would average 7 / 5 = +1.4 per deck. If we then shuffle our cut-off slug with one of the discard decks, we would estimate a count of -7 + 1.4 = -5.6 for the new two-deck shuffled slug. We could then cut these cards to the bottom of the shoe, adjust our starting running count, and play with a significant advantage in a four-deck game. We are essentially using the cut card to "short the deck" of cards we don't want. Similarly, this method can also be used to cut good cards do the top of the shoe.
As we can see, this can be an incredibly powerful tool to use in actual casino play. Unfortunately, the computations can be a bit too clumsy for some of us to have ready when the cut card lands in front of us. Hence the need for a system that can be employed by the average imbecile.
The Approximation Formula
So isn't there an easier way to get from point A to point B? Happily, yes! Let's take a look at the formula we have already:
Shuffled slug = cut-off slug + average count density (per slug)
Average count density = discards / (number of slugs in shoe - 1)
To break this formula down, we will see that the shuffled slug (the value of the cut-off part that we are tracking AFTER the shuffle) equals the original value of the cut-off slug plus the value of the estimated average count of each slug from the discards. The average count is found by dividing the known count (the running count before the shuffle) by the number of slugs it is comprised of (number of slugs - 1). We subtract 1 because we don't want to include the cut-off slug in our division because it has it's own value already.
This is the standard formula which most of you have probably wrestled with while the dealer is shuffling and stacking away. Although it is very straightforward, the division to find the average count density can be difficult when awkward numbers are used. How many of you would have come up with +1.4 in the above example? After a few hours of casino practice, I decided that I couldn't get it. I was having problems with switching the signs as well. I would get confused with the "negative slugs are GOOD now" concept and was afraid that I would cut a bad slug to the front by accident. So I did what anyone with the mentality of a thirteen-year-old boy would do: I whined about it being "too hard" and gave up.
A few months later I sat down with Excel and used the above formula to make a spreadsheet showing different running counts for the cut-off slug and their final outcomes. I thought that having the formula with various solutions in front of me would help me to understand the concepts and perhaps memorize some of the tricky division problems. I figured that memorizing +1.4 is easier than finding 7 / 5. However, after staring at the numbers for a while, something occurred to me. Why am I going through all of this trouble? Why am I swapping signs, subtracting slugs, and dividing "average count densities"? If I have to figure out how many 1.5 deck slugs are in a six-deck game I'll scream! Yes, I know the answer is 4 and it's easy to remember - but when you're starting out and the dealers are using different penetration levels, it can become maddening. That's when I saw the shortcut.