#1




Probs: HiLo TC=0, Single Deck
I'm not sure I should post this, but I thought some might find it interesting.
One of the programs I have written computes the probabilities of drawing any rank for either HiLo or KO given any running count at a given pen starting with from 18 decks with optionally including specific removals in the calculation. It uses a weighted average of all of the possible subsets for a given running count/pen combination. The optional removals are good for getting the insurance decision to a greater degree of accuracy. For example, single deck AA v A insure at HiLo TC >= 2.36! (yes minus.) A hand such as 79 v A would be insured single deck at a HiLo TC >= 0 wheras TT v A insured at HiLo TC >=+ 3.71. The greater number of decks, the less variation in the comp dependent index. Insurance isn't why I started this post, though. It just came out that way. It turns out that the only times the probabilities of drawing a nonten=1/13 and drawing a ten=4/13 exactly at a HiLo TC=0 for a single deck is when there are exactly 52 or 26 cards remaining to be dealt. For any number of decks, the starting probabilites of 1/13 and 4/13 only occur exactly when no cards or 1/2 the cards are removed, assuming no specific removals. What does this mean? In general, it is a little more likely than you might think that you will draw either a low card or ten/ace and less likely to draw a 79 at a HiLo TC=0 when more than half the deck remains and just the opposite when less than half the deck remains at a HiLo TC=0. When very few cards remain, the probabilities fluctuate greatly. The probabilities also fluctuate when the deck is nearly full. I'm not too good with spreadsheets, but I created one with a chart. I'll attempt to attach it to this post. The effect of the differing probs at TC= 0 probably doesn't amount to much, but I thought it may be of interest. k_c 
#2




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Anyway interesting stuff. 
#3




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Take 2 simple cases: 51 cards remaining and 1 card remaining dealt from a single deck where RC=TC=0. When there are 51 cards, the card removed has to be a 7,8, or 9 so the probability of drawing a 7,8, or 9 < 1/13. When there is 1 card remaining, that card has to be a 7,8, or 9 so the probability of drawing a 7 = prob of drawing 8 = prob of drawing 9 = 1/3 and obviously 1/3 > 1/13. To get the other points on the chart, I have a program that can compute the probabilitiies of drawing each rank given a running count and a given number of cards remaining for either HiLo or KO (I used HiLo) and can optionally account for specific removals. It's based on a weighted average of all of the possible subsets for the given count/pen combo. So far the only reasonable use I have found for it is to more accurately get insurance indices for HiLo and KO. You can get count comp dependent indices for hands of LL, LM, LT, LA, MM, MT, MA, TT, TA, and AA where L means low, M means medium (cards with 0 tag,) T means ten and A means ace. The extra accuracy would be the most worthwhile for a single deck game, declining in value as more decks are added. Anyway, I probably didn't explain it well, but hopefully the general idea can be seen. k_c 
#4




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I mean at that point, with 39 cards left, RC=TC=0, all ratios are the same, so what's the diff from 26 cards left? Anyway, it did make me wonder what count I should I use if I ever encounter a 21 BJ game. I mean we all know that assigning +1 to all cards except 10, and 2 to all 10's, that when this RC (no TC necessary) gets above 4 times the number of decks used, an insurance bet becomes profitable no matter how many decks have yet to be dealt. So it's a perfect count for insurance decisions. Of course such a count is not very useful for estimating advantage for other betting decisions. Great as a second count if you can do it lol. 
#5




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6 (26), 1 (79), 6(TA) 5 (26), 3 (79), 5 (TA) 4 (26), 5 (79), 4 (TA) 3 (26), 7 (79), 3 (TA) 2 (26), 9 (79), 2 (TA) 1 (26), 11 (79), 1 (TA) Code:
Cards in deck = 13 p(2) = 7.64734978774833E02 p(3) = 7.64734978774833E02 p(4) = 7.64734978774833E02 p(5) = 7.64734978774833E02 p(6) = 7.64734978774833E02 p(7) = 7.84216737417225E02 p(8) = 7.84216737417225E02 p(9) = 7.84216737417225E02 p(10) = 0.305893991509933 p(1) = 7.64734978774833E02 Code:
Cards in deck = 52 p(2) = 7.69230769230769E02 p(3) = 7.69230769230769E02 p(4) = 7.69230769230769E02 p(5) = 7.69230769230769E02 p(6) = 7.69230769230769E02 p(7) = 7.69230769230769E02 p(8) = 7.69230769230769E02 p(9) = 7.69230769230769E02 p(10) = 0.307692307692308 p(1) = 7.69230769230769E02 I'm sorry for all the decimal places and scientific format of the numbers. I just pasted them from a previous output file. Quote:
k_c 
#6




Me either
Well maybe I only meant that it's possible that the original deck probabilities could exist at 39 and 13 cards lol. Not that they must, if that was your point. What does your program say if the 1st 13 cards are all spades? What does your program say if the 1st 12 cards are all the 7,8,9's followed by 4 2's, 4 10's, 3 3's and 3 J's? 26 cards left, RC=TC=0 but no 2's or 10's left? Must be that weighted thing. Anyway no big deal. I have no idea what I'm talking about anyway. No surprise there. 
#7




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Cards in deck=26 (TC=00.0, ins EV=1.04%) p(2) = 0.00000 p(3) = 0.03846 p(4) = 0.15385 p(5) = 0.15385 p(6) = 0.15385 p(7) = 0.00000 p(8) = 0.00000 p(9) = 0.00000 p(10) = 0.34615 p(1) = 0.15385 Quote:
k_c 
#8




Remove all spades
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Good question . k_c 
#9




That's all I meant lol. But I wish I knew what it meant

#10




At the risk of confusing you, I'll try to reference what happens when 13 spades are removed, which you seem to understand.
When 13 spades are removed, there is 1 and only 1 HiLo composition possible: 15 (26), 9 (79), 15 (TA) When 13 cards are removed and you stipulate that running count = 0, there are 6 possible HiLo compositions possible: 14 (26), 11 (79), 14 (TA) 15 (26), 9 (79), 15 (TA) 16 (26), 7 (79), 16 (TA) 17 (26), 5 (79), 17 (TA) 18 (26), 3 (79), 18 (TA) 19 (26), 1 (79), 19 (TA) Without going into any calculations, there is more to figure out in the second case. Each composition has it's own probability of occurring and also it's own probability of drawing each rank. To get the overall probability of each rank, you give more numerical weight based on their probability to the compositions more likely to occur and less weight to the ones less likely and compute an average. In the first case you are stipulating that a specific number of each rank be removed (specific removals.) In the second case you are only stipulating that 13 cards be removed and that the running count = 0 (no specific removals.) Can you see now why it is possible to have different probabilities of drawing each rank at a HiLo RC = 0 with different pen levels? That's about as clearly as I can say it. I guess I'm crazy to go this far lol. k_c 
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