First capture by the WayBack Machine (web.archive.org) May 21, 2020.
Let me just say that benign ignorance has been at the heart of the matter. Nobody really knew what the real odds (probability) of blackjack were. Analysts lacked the fundamental elements required by the fundamental formula of probability: favorable cases (over) total possible cases.
Calculating the odds is the sine qua non condition of calculating the house advantage or the edge the casinos have in the game of blackjack. No casino offers a game where they don't have an edge or advantage. It's their bloodline — a legal requirement, as a matter of fact.
The first attempt at calculating the house advantage in blackjack is granted to John Scarne, a non-mathematical man who had the ambition of being the greatest gambling writer in history. Personally, I grant such honor to Blaise Pascal who analyzed a backgammon game. The historical event is known as de Méré Case and it founded a branch of mathematics hence known as theory of probability.
John Scarne rightly figured out that the casino gains an edge in blackjack because of the simultaneous bust — the dealer and the player bust at the same time. However, when the player busts, he/she loses the bet immediately as he/she always plays first. It is possible that the dealer can bust his/her hand (in the same round), but it is too late for the player; they already lost their bet.
John Scarne calculated the odds of dealer's bust to be 28%. If the player played by the same rules as the dealer, the simultaneous bust would be: 0.28 * 0.28 = 7.8%. But since the player is allowed to stand on 16 or less under certain circumstances, our "mafia" man calculated that the final odds would be around 5.9%. That's the "physical probability" of casino winning at blackjack.
The casino offers bonuses to the player, however. They pay 3 to 2 for a natural 21 (Ace+Ten in the first 2 hands of the player). They also allow double-down and splitting pairs. At the end of the day, the bj house advantage goes all the way down to that glamorous figure of 0.5%.
Right now, we focus our attention on the raw figure of 5.9%. Based on that figure (and so-called simulations), everybody agreed that the results of blackjack were:
In order to calculate the probability precisely, we must generate all the elements (blackjack hands) in lexicographical order. Nobody even knows how many hands are possible, as their size varies widely: From two cards to 10 cards (for one deck)! When two or more decks are employed, the blackjack hands can go from two cards to 11 cards.
Of course, there is a lot of blackjack software out there! But all that software belongs to the simulation category. That is, the blackjack hands are dealt randomly. Based on the well-known-by-now Ion Saliu's Paradox, random generation does not generate all possible combinations, as some elements repeat. So, we can never calculate the probability precisely based on random generation. If there are 334,490,044 total possible complete hands in blackjack, only 63% will be unique and 37% will be repeats — if we randomly generate 334,490,044 hands.
I had started years ago a blackjack project to generate all possible hands. It was very difficult. I found the project in the year of grace 2009 and also the code to generate sets from a list (last update: 2014). In this case, the list is a 52-line text file with the values of the blackjack cards, from the four 2's to the 16 Tens, to the four Aces. That's a stringent mathematical requirement. The deck of cards must be also ordered lexicographically, if we want to correctly generate all qualified sets in lexicographical order.
I generated blackjack hands as both combinations and arrangements. Then, I opened the output files (text format) and checked as many hands as possible. Yes, computing things are so much better today than just a decade ago. The generating process is significantly faster.
I wrote a special Web page dedicated to the topic of calculating precisely mathematically the bust-odds at blackjack following the Dealer's rules. There are lots of details, plus screenshots of the probability programs:
Keep this new figure in mind: The odds for a blackjack Dealer's bust are at least 33%. The bust probability is calculated by dividing the number of Dealer's busted hands to the total possible blackjack actions. Blackjack actions is a parameter that counts everything: Busted hands, pat hands (17 to 21), blackjack hands, and draws or hits to the first 2-card hands (incomplete hands). The software does NOT print the incomplete bj hands.
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 16 and under, stand on all 17 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.
I believe that setting the hit limit to 14 or 13 reflects pretty closely 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%
Recalculating the raw figures for winning/losing hands, my theory shows:
Axiomatic ones, who's right and who's wrong? If you have been a frequent visitor of my website, you already know how many hits I've been taken from casino executives, agents, moles, other gambling authors, system developers, vendors, gurus, bishops, saints, etc. Granted, the attacks against yours truly were far more intense earlier (beginning 1998 and ending early 2000's). They realized I wouldn't get intimidated, so they have given up, by and large.
In this year of grace 2019, I came up with a new idea: Let's set at the same table mathematics and reality. The first attacks aginst me went along the lines: "Mathematics, specifically formulae, have no place in gambling — as it is totally random." And I've always counterattacked: "But what is not random, crooked idiots? The entire Universe is ruled by Almighty Randomness, as voided of consciousness as it might be!"
Standard deviation is the watchdog of randomness. Let's see what figures of blackjack odds are right by employing the binomial standard deviation. Then, compare the results to casino gambling reality.
It is time now to apply the most important bonuses the casinos grant to the blackjack players:
We ignore the current tendency in the gambling industry to pay a natural bj 6-to-5.
The double down success is closely around 60%. The same success rate of 60% occurs in the pair splitting situations.
Next, it is very important to know the probability/odds of appearance for the 3 bonuses above.
Axiomatics, we run my probability software widely known as SuperFormula.exe, the function D: Standard Deviation. We run the function twice: First, for the traditional black jack parameters (5.9% odds, 48% winning probability for the player); secondly, for what I consider closer-to-reality blackjack parameters.
We take a common case of playing 100 hands. That is, the blackjack player must cash in the amount needed to play 100 hands at the minimum bet. For example, in the rare case of $10 minimum bet, the player must chip in at least $1000. I can't stress enough the stupidity of players who start with $100... they lose quickly... then leave the table... go to another table and cash in $100... etc. Vae victis! Poor victims!
The standard deviation for an event of probability
p = .478
in 100 binomial experiments is:
BSD = 5
The expected (theoretical) number of successes is: 48
Based on the Normal Probability Rule:
• 68.2% of the successes will fall within 1 Standard Deviation
from 48 - i.e., between 43 - 53
•• 95.4% of the successes will fall within 2 Standard Deviations
from 48 - i.e., between 38 - 58
••• 99.7% of the successes will fall within 3 Standard Deviations
from 48 - i.e., between 33 - 63
Without bonuses (BJ=1.5, double-down=2, split-pairs=2)
52 - 48 = 4 hands net loss after 100 hands; 40 units for
betting unit = 10 (dollars, etc.)
With bonuses
A1. 4% BJ pays 1.5 to 1
48 winning hands * 4% = 1.9
1.9 * 1.5 = 2.9 extra hands (added to the average of 48)
B1. Double down hands: 8%
48 winning hands * 8% = 3.8
Double down hands win 60% of the time: 2.3
2.3 * 2 = 4.6 extra hands (added to the average of 48)
C1. Pair Splitting hands: 3%
48 winning hands * 3% = 1.44
Split hands win 60% of the time: 1.35 * 0.6 = 0.86
0.86 * 2 = 1.7 extra hand (added to the average of 48)
Total bonus hands
48 + 2.9 + 4.6 + 1.7 = 57.2
Recalculated net loss
52 - 57.2 = 5.2 hands NET WIN after 100 hands; 52 units for
betting unit = 10 (dollars, etc.)
WHOA! ON AVERAGE, THE PLAYER WINS 52 BET UNITS AFTER PLAYING 100 HANDS!!! That's a flagrant impossibility in 99.7% to all blackjack players, in all casino situations. You and I will never, ever, see a basic strategy player be ahead $52 after playing 100 hands, at $10 table minimum!
We come back to earth by going with my fundamental blackjack parameter: 45% winning odds for the player.
The standard deviation for an event of probability
p = .45
in 100 binomial experiments is:
BSD = 4.97
The expected (theoretical) number of successes is: 45
Based on the Normal Probability Rule:
• 68.2% of the successes will fall within 1 Standard Deviation
from 45 - i.e., between 40 - 50
•• 95.4% of the successes will fall within 2 Standard Deviations
from 45 - i.e., between 35 - 55
••• 99.7% of the successes will fall within 3 Standard Deviations
from 45 - i.e., between 30 - 60
Without bonuses (BJ=1.5, double-down=2, split-pairs=2)
55 - 45 = 10 hands net loss after 100 hands; 100 units for
betting unit = 10 (dollars, etc.)
With bonuses
A2. 4% BJ pays 1.5 to 1
45 winning hands * 4% = 1.8
1.8 * 1.5 = 2.7 extra hands (added to the average of 45)
B2. Double down hands: 8%
45 winning hands * 8% = 3.6
Double down hands win 60% of the time: 2.2
2.2 * 2 = 4.4 extra hands (added to the average of 45)
C2. Pair Splitting hands: 3%
45 winning hands * 3% = 1.35
Split hands win 60% of the time: 1.35 * 0.6 = 0.8
0.8 * 2 = 1.6 extra hand (added to the average of 45)
Total bonus hands
45 + 2.7 + 4.4 + 1.6 = 53.7
Recalculated net loss
55 - 53.7 = 1.3 hands net loss after 100 hands; 13 units total loss for
betting unit = 10 (dollars, etc.)
You, the player, do lose. Still, this is the happiest case calculated by my blackjack-odds software: One deck of cards. Today's PCs are still incapable (at least in the case of this programmer) to calculate for two or more decks of cards. But I experimented with calculable amounts of cards. The rule is very clear: The more cards, the worst the odds get for the player. In other words, the more decks, the worse conditions for the blackjack hopeful! And even worse with multiple players at the table (the common reality)!
Haven't you witnessed this in any casino, at any blackjack table? The overwhelmingly vast majority of players lose their bankroll quickly. They leave the venues almost on their knees. "How the hell is this possible," they ask themselves (sometimes loudly). "Blackjack is supposed to be a 50-50 game... damn it!"
It ain't such a golden coin game, kokodrilo (royalty-name for big-time gambler)! I'm afraid you were misguided big-time... you still are. You are mostly cheated by the card-counting crooks, the bedfellows of the casinos in that gambling bedlam! You go by their insane odds and you are guaranteed to win as a matter of fact. Play 100 hands and win $52 at $10 minimum bet. Well, then, ask for a $100 table minimum and make a $500 net. This is the average, but it will be confirmed in any reasonable long run. Not the billions of hands long-run prophesized by the crooks!
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