Strong mathematics follows. STRONG == STill wRONG
The miscounting has been moved (or duplicated) to: http://saliu.com/blackjack.html
"How can we apply the new programming to determine the bust odds for the blackjack Player? After heated debates in forums in 2014, I simply modified my software. The hit-stand limits can be set by the user. Initially, it was fixed -- the ubiquitous hit all 16s and under, stand on all 17s or greater.
"The software user can set the hit-limit to any value. The choices are, obviously, from 12 to 16. I tried, for example, the hit limit to 11 -- that is, hit anything 11 or under, stand on anything 12 or higher. Evidently, there is no bust in such situations. That's another proof that my programming is 100% correct. [100% correct programming of inappropriate algorithms for a problem produces what?]
"I believe that setting the hit limit to 14 or 13 reflects pretty closely ["mathematical" hand-waving!!!] the bust odds for the Player. That is, stand on 15 or greater (as arrangements):
"Percentage Player Bust: 61656 / 274254 = 22.48%
"Or, stand on 14 or greater (as arrangements):
"Percentage Player Bust: 570 / 3702 = 15.40%
"Now, the house edge goes between something like .3355 * .2248 = 8.3% and something like .3355 * .1978 = 6.6%. It averages out to 7.5%. [Using the mean, a mathematical method, must mean it is correct, right?] It is a far cry from the intentionally false house advantage (HA) of 1%, or even .5%! The overwhelming majority of blackjack players lose their bankrolls quickly, because this is NOT a 50-50 game or so much close to that margin! And always be mindful that blackjack is strongly sequential: The Dealer always plays the last hand. Otherwise, the casinos would go bankrupt!"
At least Lord Parpaluck is using "arrangments" instead of "combinations." However, the counting of arrangements to compute likelihoods (probabilities) requires that each arrangement is equally likely. In order for this to occur, one needs to make the number of cards in the arrangement to be the same in every case. Collapsing arrangements into equivalence classes and then counting classes is incorrect mathematics. Lord Parpaluck needs to lexicographically generate arrangements (sequences) of equal lengths. Then, the results will match those found at: http://wizardofodds.com/games/blackjack/appendix/2b/
If they are not part of the "formula", how does the calculation of HA take into account a player's ability to split, double down, and receive 3:2 for a natural? Of course, the game rules such as h17 or s17 are not mentioned either.
Now, let us assume that Parpaluck's simple formula is (basically) correct. The blackjack house edge is equal to .3355 * player-bust-likelihood. ".3355" is the likelihood of dealer busting. Let us suppose that we change the blackjack rules so that the dealer never busts. So, any 22, 23, ..., 26 for the dealer is treated as if it were a 17. Would you like to play this game of blackjack? So, instead of ".3355", ".0000" should be used in Parpaluck's blackjack HA formula. The house advantage then goes between something like .0000 * .2248 = 0.0% and something like .0000 * .1978 = 0.0%. This axiomatic mathematics shows that "no dealer bust blackjack" is a bad game for a casino and good for its customers. Instead, casinos should change the blackjack rules so that the dealer always draws until reaching 22 or more, insuring that the dealer always busts. Then, the house advantage goes between something like 1.0000 * .2248 = 22.5% and something like 1.0000 * .1978 = 19.8%. The blackjack tables should then become about three times as profitable, until all of the customers realize how bad the game is and stop playing.
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