Where did you play it and how did you verify it's countable?Originally Posted by aceside
Let’s wait for K_C and Cacarulo’s results first.
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aceside,
The first graph of this post answers your queries:
https://www.blackjacktheforum.com/sh...ncy-vs-HiLo-TC
Hope this helps!
Dog Hand
Your results are from CVData simulation. They are accurate enough, but I still hope to find a theoretical calculation. Furthermore, I specify the dealing depth at this particular value 208/312 to find out if theoretical values match your simulation results or not.
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I actually have calculated the first graph. My calculation result for the probability of a dealer's ace up card is P=9.6% (at TC=+10 when 208 cards were dealt out from a 6-deck shoe) matches perfectly to Dog Hand's result in the first graph! Wonderful!
Note: I revised my numbers.
Last edited by aceside; 06-05-2022 at 12:01 PM.
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Dog Hand results are correct! I've double checked my math.
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aceside,
One difference is that you are calculating the percentages specifically at 208 cards penetration, while my graphs are a weighted averaged over all penetrations up to 75% (actually, a bit under 234 cards, since the cut card appears during the last round).
The case you showed as (35, 24, 45) corresponds to 43.27% bigs, but the (small, middle, big) distribution will vary slightly with penetration: for example, if your penetration is 10 (instead of 208), the only possible +10 distribution is (110, 72, 120) or 39.74% bigs, since all 10 cards have to be small. On the other hand, if the penetration is 302, the only possible +10 distribution is (0, 0, 10), or 100% bigs... Hint, bet the MAX! ;-)
Hope this helps!
Dog Hand
I miscalculated the TC in my first version and realized my mistake. Thank you for helping!
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If 208 cards are randomly dealt and counted from a 6 deck shoe, running count is +10, and no other cards are specifically removed the probabilities of each up card would be p[1] through p[10] below.
An estimate of dealer probabilities based on dealing from a shoe using those rank probabilities for both dealer standing on and hitting soft 17 are below.Code:Count tags {1,-1,-1,-1,-1,-1,0,0,0,1} Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0} No subgroup (removals) are defined Decks: 6 (possible input for cards remaining: 1 to 312) Cards remaining before up card (current = 156, no input = no change): 104 Initial running count (full shoe): 0 Running count (before up card is dealt, no input defaults to 0): 10 Computing.....please wait..... Number of subsets for above conditions: 37 Prob of running count 10 with above removals from 6 decks: 0.02147 p[1] 0.08657 p[2] 0.06734 p[3] 0.06734 p[4] 0.06734 p[5] 0.06734 p[6] 0.06734 p[7] 0.07682 p[8] 0.07682 p[9] 0.07682 p[10] 0.34628
k_cCode:Dealer probabilities - 6 decks dealer stands on soft 17.... Count tags {1,-1,-1,-1,-1,-1,0,0,0,1} Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0} Subgroup removals: None Running count (before up card is dealt): 10 Cards remaining: 104 Up card 17 18 19 20 21 BJ BUST 1 0.1125 0.1217 0.1223 0.1243 0.0474 0.3496 0.1222 2 0.1288 0.1221 0.1267 0.1230 0.1198 0.0000 0.3795 3 0.1221 0.1196 0.1155 0.1199 0.1165 0.0000 0.4065 4 0.1205 0.1086 0.1117 0.1093 0.1133 0.0000 0.4366 5 0.1125 0.1119 0.1081 0.1007 0.1015 0.0000 0.4653 6 0.1714 0.0951 0.0969 0.0930 0.0909 0.0000 0.4528 7 0.4002 0.1369 0.0683 0.0705 0.0666 0.0000 0.2574 8 0.1209 0.3903 0.1295 0.0609 0.0631 0.0000 0.2353 9 0.1131 0.1027 0.3846 0.1232 0.0546 0.0000 0.2218 10 0.1062 0.1038 0.1052 0.3677 0.0292 0.0874 0.2004 Overall 0.1394 0.1324 0.1294 0.1944 0.0649 0.0605 0.2790 Press c or C for values conditioned on no dealer BJ, any other key to exit Dealer probabilities - 6 decks dealer hits soft 17.... Count tags {1,-1,-1,-1,-1,-1,0,0,0,1} Specific removals (1 - 10): {0,0,0,0,0,0,0,0,0,0} Subgroup removals: None Running count (before up card is dealt): 10 Cards remaining: 104 Up card 17 18 19 20 21 BJ BUST 1 0.0535 0.1317 0.1315 0.1336 0.0567 0.3496 0.1433 2 0.1200 0.1237 0.1280 0.1244 0.1212 0.0000 0.3827 3 0.1149 0.1208 0.1166 0.1209 0.1176 0.0000 0.4092 4 0.1128 0.1100 0.1128 0.1105 0.1144 0.0000 0.4395 5 0.1083 0.1126 0.1087 0.1014 0.1022 0.0000 0.4669 6 0.1174 0.1044 0.1053 0.1015 0.0995 0.0000 0.4719 7 0.4002 0.1369 0.0683 0.0705 0.0666 0.0000 0.2574 8 0.1209 0.3903 0.1295 0.0609 0.0631 0.0000 0.2353 9 0.1131 0.1027 0.3846 0.1232 0.0546 0.0000 0.2218 10 0.1062 0.1038 0.1052 0.3677 0.0292 0.0874 0.2004 Overall 0.1287 0.1342 0.1310 0.1961 0.0666 0.0605 0.2828 Press c or C for values conditioned on no dealer BJ, any other key to exit
This work is done magically! I’ve been waiting for this for a long time. Thank you for your hard work! For the sake of the Bust Bonus side bet, let me focus on the hit-17 game and do a little additional calculation based on your results:
When the dealer shows an Ace up card, she will peek for blackjack. After this peeking, the dealer bust rate becomes
0.1433/(1-0.34628)=0.2192.
When the dealer shows a ten-valued up card, she will peek for blackjack. After this peeking, the dealer bust rate becomes
0.2004/(1-0.08657)=0.2194.
These two numbers look good, but there is a tiny math problem here. For example, when the dealer shows an Ace up card, the probability of her having a ten-valued under card is slightly greater than 0.34628 though, and thus the probability of having a blackjack is more. Can you also help with this part of calculation?
P.S. I thought about this again and realized that it was an overkill to consider this tiny math problem. In the subsequent calculation, we just believe the remaining 104-card deck is an infinite deck with a constant probability of each existing card as you calculated. For this side bet, what really matters is the dealer bust rate after peeking.
Last edited by aceside; 06-06-2022 at 09:22 AM.
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