Below, Paddyboy expressed the understandable feeling that the NRS formula seems counter-intuitive. What follows is an attempt to explain in simple terms the basic reasoning underlying that very useful tool for shuffle trackers. Nothing presented here is original. The ideas are developed in more sophisticated form in posts by Statman and DvBj, archived at bjmath.com in the section entitled "Shuffle Tracking / Card Location", which is accessible from that site's Table of Contents page.
Consider the first deck of an 8-deck shoe and imagine the following two scenarios.
Scenario 1. Before the shoe commences, you happen to know that the first deck has a total hi-lo running count of -4. That is, it is a favourable deck, having 4 more big than small cards.
Scenario 2. Before the shoe commences, you happen to know that the first deck is made up of a half-deck, X, with a count of -4, that is mixed with another half-deck, Y, with an unknown count. That is, you know the count of X, but know nothing specific about Y.
After the first round of play, you observe that the running count has dropped to -4.
Is your expectation for the next round better in scenario 1 or scenario 2?
The NRS formula tells us that you are better off in scenario 2.
Let's see why.
In scenario 1 you know that the count of the cards remaining in the first deck must be 0. This is because the overall count for the first deck is -4, and part way through the first deck the running count has already reached this end total. In other words, you know that 4 more high than low cards are due to come out during the first deck, and this has already happened by the end of the first round. There is no longer an excess of high cards for the remainder of the first deck.
Scenario 2 is different. Here we don't know for sure what the overall count is for the first deck. We know that one half of it, X, has 4 more big than small cards. But we don't know anything directly about the other half deck, Y. So the question arises: is the running count after the first round -4 because the extra big cards from X have been dealt, or because the unknown half deck, Y, just so happened to have an excess of big cards as well?
The correct answer to this question (which is what the NRS formula achieves) involves balancing two different considerations:
CA. Prior to the commencement of the shoe, we could form our 'best guess' of the first deck's count. Since the first deck is made up of 26 cards with a count of -4, and another 26 cards drawn from 7.5 decks with a count of +4, our best guess would be -4 + 4/7.5 = -3.47.
But there is another, separate issue to consider.
CB. In the dealing of the first round, the dealer has drawn cards that have probably come partly from the known half-deck X and partly from the unknown half-deck Y. Perhaps the negative count at the end of the first round is because cards were mainly drawn from X, which we know has a negative count. On the other hand, perhaps many of the cards didn't come from X, but instead came from Y. We don't know much about Y. Our best guess could easily be wrong. Therefore, perhaps the negative count of the first round provides us with some evidence that Y also contains an excess of big cards, just like X.
The idea behind CB might be made clearer by a simple example. Suppose I say to you, "I am holding in my hand 2 half decks, A and B. A has a count of +10, B has a count of -10. I will deal 5 cards from one of the half decks. Then you must guess whether I am dealing from A or B." Imagine that I deal the 5 cards and the count goes negative. Would you guess that I am dealing from A or B? The correct answer could be either, but your best guess would certainly be B. A small sample from B is more likely to give a negative count than a small sample from A, simply because on average our sample will reflect its population.
To sum up, our prior knowledge (before the shoe commences) suggests that Y will have a slightly positive count. This is CA. But our sampling evidence (once a round has been dealt) suggests that Y may actually have a negative count. This is CB. By correctly weighting these two considerations, the NRS formula is able to provide a correct answer for any particular situation.
Relating all this back to our 8-deck game with 1 round dealt and a running count of -4, the critical point is that we don't know where the 4 excess big cards came from. Perhaps they came from X. But they also may have come from Y, which would mean that the extra big cards that we know are in X would still remain to be dealt.
The implications for the player's expectation over the first deck are quite interesting:
IA. Prior to the commencement of the shoe, the player's expectation for the first deck is higher in scenario 1 than scenario 2. This is simply because the first deck is expected to have a count of -4 in scenario 1, but -3.47 in scenario 2. Thus, the expected excess of big cards is slightly greater in scenario 1 than scenario 2.
However...
IB. After one round has been dealt and the running count is -4, the player's expectation for the remainder of the first deck is now higher in scenario 2 than in scenario 1. The situation has been reversed. In scenario 1 no excess of big cards remains, whereas in scenario 2 the running count will still be expected to fall.
If the basic tension between CA and CB is kept in mind, perhaps IA and IB will not seem quite so counter-intuitive.
Consider the first deck of an 8-deck shoe and imagine the following two scenarios.
Scenario 1. Before the shoe commences, you happen to know that the first deck has a total hi-lo running count of -4. That is, it is a favourable deck, having 4 more big than small cards.
Scenario 2. Before the shoe commences, you happen to know that the first deck is made up of a half-deck, X, with a count of -4, that is mixed with another half-deck, Y, with an unknown count. That is, you know the count of X, but know nothing specific about Y.
After the first round of play, you observe that the running count has dropped to -4.
Is your expectation for the next round better in scenario 1 or scenario 2?
The NRS formula tells us that you are better off in scenario 2.
Let's see why.
In scenario 1 you know that the count of the cards remaining in the first deck must be 0. This is because the overall count for the first deck is -4, and part way through the first deck the running count has already reached this end total. In other words, you know that 4 more high than low cards are due to come out during the first deck, and this has already happened by the end of the first round. There is no longer an excess of high cards for the remainder of the first deck.
Scenario 2 is different. Here we don't know for sure what the overall count is for the first deck. We know that one half of it, X, has 4 more big than small cards. But we don't know anything directly about the other half deck, Y. So the question arises: is the running count after the first round -4 because the extra big cards from X have been dealt, or because the unknown half deck, Y, just so happened to have an excess of big cards as well?
The correct answer to this question (which is what the NRS formula achieves) involves balancing two different considerations:
CA. Prior to the commencement of the shoe, we could form our 'best guess' of the first deck's count. Since the first deck is made up of 26 cards with a count of -4, and another 26 cards drawn from 7.5 decks with a count of +4, our best guess would be -4 + 4/7.5 = -3.47.
But there is another, separate issue to consider.
CB. In the dealing of the first round, the dealer has drawn cards that have probably come partly from the known half-deck X and partly from the unknown half-deck Y. Perhaps the negative count at the end of the first round is because cards were mainly drawn from X, which we know has a negative count. On the other hand, perhaps many of the cards didn't come from X, but instead came from Y. We don't know much about Y. Our best guess could easily be wrong. Therefore, perhaps the negative count of the first round provides us with some evidence that Y also contains an excess of big cards, just like X.
The idea behind CB might be made clearer by a simple example. Suppose I say to you, "I am holding in my hand 2 half decks, A and B. A has a count of +10, B has a count of -10. I will deal 5 cards from one of the half decks. Then you must guess whether I am dealing from A or B." Imagine that I deal the 5 cards and the count goes negative. Would you guess that I am dealing from A or B? The correct answer could be either, but your best guess would certainly be B. A small sample from B is more likely to give a negative count than a small sample from A, simply because on average our sample will reflect its population.
To sum up, our prior knowledge (before the shoe commences) suggests that Y will have a slightly positive count. This is CA. But our sampling evidence (once a round has been dealt) suggests that Y may actually have a negative count. This is CB. By correctly weighting these two considerations, the NRS formula is able to provide a correct answer for any particular situation.
Relating all this back to our 8-deck game with 1 round dealt and a running count of -4, the critical point is that we don't know where the 4 excess big cards came from. Perhaps they came from X. But they also may have come from Y, which would mean that the extra big cards that we know are in X would still remain to be dealt.
The implications for the player's expectation over the first deck are quite interesting:
IA. Prior to the commencement of the shoe, the player's expectation for the first deck is higher in scenario 1 than scenario 2. This is simply because the first deck is expected to have a count of -4 in scenario 1, but -3.47 in scenario 2. Thus, the expected excess of big cards is slightly greater in scenario 1 than scenario 2.
However...
IB. After one round has been dealt and the running count is -4, the player's expectation for the remainder of the first deck is now higher in scenario 2 than in scenario 1. The situation has been reversed. In scenario 1 no excess of big cards remains, whereas in scenario 2 the running count will still be expected to fall.
If the basic tension between CA and CB is kept in mind, perhaps IA and IB will not seem quite so counter-intuitive.
