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#1
March 22nd, 2009, 03:22 PM
 stophon Senior Member Join Date: Feb 2009 Posts: 141
Ace Tracking Methodology

Alright so I'm one of those guys who likes to figure stuff out for myself, so here was my attempt at solving Ace Tracking. Please check my methodology.

An ace yields a 51% advantage if it is your first card.
Stophon shuffling data:
Clumps of one: 77.6%
Clumps of two: 20.0%
Clumps of three: 2.4%
Clumps of four or more: 0%

(This data matches up pretty well with some of the ranges Snyder gives on his site for shuffles. I ran 5 "trials" to get those numbers, I'm sure that in a casino you will get 4+ more often, however since the numbers will be very small it shouldn't affect betting that much I don't think.)

Possible ways a key card can occur
K-Key card, O-Other Card, A-Ace
Original Setup
O--------- O
K---- +---- O
A--------- O
O--------- O

Arrangement 1
K--------
----- O-- = P(1 card)*P(A&K not grouped)
A-------- = (.776)(.776) = 60.2%

Arrangement2
K----------
----- O---- = (P 2 cards)*(P K&A not grouped)
----- O---- =(.20)(.776) = 15.5%
A----------

Arrangement3
K --------- = P(grouped) = P(2 cards) + P(3 cards)
A----------= (.20)+(.024) = 22.4%

Arrangment4
K-----------
----- O---- =P(3 cards)*P(not grouped)
----- O---- = (.024)(.776) = 1.9%
----- O----
A----------

(It added up to 100%! But, that doesn't by any means prove that my math was correct. Please point out anything if it is wrong.)

So all of this means that the probability of:
k , A------------22.4%
k , O, A---------60.2%
k, O, O, A-------15.5%
k, O,O,O,A-------1.9%

So our betting plan to get a fully Kelly bet:
f* = Advantage * P(Ace in player's hand)
so if you see the key card come out as the last card then
f* = 0.51 * 0.621
f* = .317
So to place a full Kelly bet, you would play 31.7% of your bankroll. However this assumes that you will get the same distribution throughout each shuffle. Now how the hell do I acquire the standard deviation charts for a shuffle? (Edit: No it does not assume the same distribution through each shuffle. It just assumes the shuffler on average will drop the same percentage of 1, 2, and 3 card clumps. That good because it means figuring out my E.V. will be much easier.)

P.S. I haven't read tracking books yet, though I searched the forum for all it had and read a bunch of online articles.

Last edited by stophon; March 22nd, 2009 at 04:25 PM. Reason: add P.S., fixed betting math