Help with Ace tracking, please check my formula

To turn Ace tracking into a complete science, we can accurately calculate the effects of the Ace in "Ace location" to alter house edge.
Hence use Ace tracking in combination with straight counting.

Let x= %edge for being dealt Ace, from theory of blackjack

j = current edge from true count-edge from Ace

f = your edge for dealer having an Ace

y = probability of Ace landing appearing in this round, can be calculated using simulations of shuffles by dealers of different consistency.

u = variance for counting edge

d = variance of Ace location

g = probability of Ace appearing in dealer's hand.

h = probability of Ace landing on your hand

i = number of hands you play/number of hands you play+1

k = 1/total number of hands

z = Additional edge gain by Ace, excluding count information.

Hence z = y(i*h*x)-(k*g*f))

So edge using true count and Ace location is:
z - i*coefficient correlation of hands*j

d = ((i*h*x)^2-z)+((k*g*f)-z)^2

Using location information for kelly betting:

z/d* br

Using counting information and Ace location

(z/d+j/u)*br

Seems okay to me. Nice work.
Minor pedantic point: the card distribution is non-random by definition when location play is possible. I don't think you can get that in a formula.

Simply calculate probability of Ace appearing with 1,2,3,4,5,6,7, cards dealt out up to 16 maximum possible in one round initially.
Then calculate mean using these probability of Ace appearance and you can even calculate the edge for each of the 7 boxes you play.

This let you bet huge variations in 1 round adding cover.

...more that the pick & pay styles of ordering discards mean that the chance of a chance following another card are not quite what you might expect, for example aces dislike other aces because they tend to get split and put into the discard tray separately. Wong's Professional Blackjack has a lot of data on this.

1) What is the significance of "number of hands you play +1" in the definition of i?

As you may know, you can't simply add together the values of more than one hand getting an ace. (Eg. If there's a 50% EV for getting an ace, and a 50% chance it will fall on hand 1 or hand 2 of two simultaneous hands, the overall EV isn't 50%x50% + 50%x50% = 2x50%x50% = 50%. Reason being that if hand 1 gets the ace, hand 2 can not, and must be penalized for that.) So I can see why you don't just multiply (#hands)*h*x, but how does dividing by (#hands +1) solve the problem?

2) Why are you multiplying by y in the first half of the formula for z? The probability, h, of an ace landing in the hand is already there. What would y*h represent? These aren't independant, unless I'm misreading your definitions. What allows you to multiply these dependant probabilities?

I think this is a very difficult problem. Can you point me to a derivation of these formulae?

ETF

Can you explain?

There is a very broad series of independant variables for the effect of how much information you have about the section of cards that has the ace, and what the boundaries of that section are, and the distribution of normal error for the location of that tracked/keyed ace.

Here is one example of this problem:

What curve describes the probability of where that ace will be dealt? Is it 50/50 one of 2 spots; a T distribution for 3 spots; a T=n distribution for falling absolutely within n spots; a normal distribution falling within a shuffle tracked subdeck that is 30 cards, or more; or the entire normal distribution of the entire shoe?

Here is the inverse of this example:

To what extent does the probability of the ace arriving at a precise spot mean that that ace is less likely to arrive at other spots? Is it one ace removed from 2 cards to one ace removed from n cards etc.?

This formula is highly incomplete!

If possible with the required skill, one would need to be playing head to head or with a team that can cover all 7 spots. There are so many variables it becomes (almost) impossible to force a final result that would stand up under fire. Having said all of that, what we do know is that there is a very nice edge to these plays, and without solid mathematical data, ie variables given a true number, we are forced to rigafratch using our own understanding of BR vs EV and our own ability. It is also good to know that we have come to the end of our ability mathematically speaking, but that never means much more to me than that we just don't know how big our edge really is....smile

Rob McGarvey

I didn't see anything about the edge for a hand if you know the first card WON'T be an ace. The average edge of a hand starting with a non-ace is about -4%.

I didn't see anything about the edge for a hand if you know the first card WON'T be an ace. The average edge of a hand starting with a non-ace is about -4%.

Could you expand on this? How could you get (partial?) information on this from sequencing? My understanding is that you can predict the appearance of aces but not always do the opposite-due to false keys, stripping reversing the order etc.