# THEORY OF PROBABILITY APPLIED TO BLACKJACK On Mathematics, True Odds, Basic Strategy, Card Counting, House Advantage, Tables

## By Ion Saliu, Blackjackologist At-Large

### I. The Fundamental Probability Issue: The True Odds at Blackjack; New Software Calculator II. The Fundamental Myth of Blackjack Gambling: Card-Counting III. The Theory of Streaks: Foundation of my Blackjack Gambling Strategy IV. Blackjack Links and Resources at This Web Site

1. The Fundamental Probability Issue: The True Odds at Blackjack - New Software Calculator

They say roulette is the queen of casino games. Then, blackjack is the king of the casino. Many believe that Blackjack, or 21, or twenty-one is the most popular casino game in the world. Blackjack is also the most researched game ever. It is also the only casino game with fluctuating odds (or probability). The winning chance changes with the composition of the deck. This is more about blackjack mathematics than anything else. For gambling is such a mathematical phenomenon that the casinos would do ... you know what to you ... if you knew well about it! This casino game is so easy to win ... but it is NOT about card counting! In truth, card counting at blackjack lopsidedly serves the greedy interests of the casinos. The blackjack player is honestly served only by the Fundamental Formula of Gambling (FFG). And thus the casino henchmen will threaten you if you simply write down in a notebook what you lost and what you won. Hey, that's a tax requirement in any jurisdiction!

Let me start by saying that the game of blackjack has caused me the most serious problems with casinos and gambling developers/authors/system vendors. Blackjack or twenty-one (seen the movie 21?) is the most popular casino game and the most researched one. There are plenty of books dedicated to the so-called mathematics of blackjack.

There is worthiness in a few of such books or eBooks. For the most part, however, there isn't much mathematics in all those blackjack studies. The heart of the matter is a worthless concept known as card counting.

I do have a strong interest in blackjack. It is well documented at my website. As a matter of fact, I consider myself the best blackjack player ever. As Muhammad Ali put: "It ain't bragging if you back it!" So, I put the money where my mouth is: I issued a casino gambling challenge, especially at the blackjack tables. So far, nobody has dared to honor my challenge. The real casino challenge is open to any gambler, gambling author, or gambling system developer — card-counting or not.

I wrote a book about the true mathematics of blackjack, insofar as precise probability calculations are concerned. You might be shocked to hear, but the mathematical truth is that your knowledge of blackjack probabilities or odds is dead wrong. Everything you had known was based on guesswork, albeit it educated guesswork.

To this date, the blackjack odds are the same as John Scarne calculated them in the 1950s. The computers were not the commodity they are today. And John Scarne was not a computer programmer! The way he calculated the odds made sense for the first two and three blackjack cards in a round. I quote from his "Scarne's New Complete Guide to Gambling" (pg. 363):

"We find that the dealer's first two cards can produce the counts from 2 to 21 in 1,326 ways."

Indeed, Combinations C (52, 2) = 1,326 two-card blackjack hands (combinations of 52 cards taken 2 at a time). That is the only thing ... half way mathematically correct! The truly correct method applies the mathematics of combinatorics alright. But instead of the numerical sets known as combinations, we must apply the mathematics of arrangements. The combinations represent boxed arrangements. In this case, C (52, 2) = (52 * 51) / 2 = 1,326. Arrangements A (52, 2) = (52 * 51) = 2,652 — or double the amount of combinations. Hence, we played cute and said half-true for the blackjack combinations case!

"We'll discover that we need to know, however, and avoid most of the fractions, if we multiply 1,326 x 169 to get a common multiple of 224,094."

Now, that's a big mystery! How did Scarne come up with that 169 factor??? Well, that's what they call an educated guess, or guestimation! John Scarne, requiescat in pace, didn't have a clue, mathematically speaking. He has never explained how he came up with that 169, kind of a new number of the beast!

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 simulator 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 rolled up my sleeves again. I had started years ago a blackjack project to generate all possible hands. It was very difficult. I found the project this year (2009) and also the code to generate sets from a list. 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.

Let's make this analogy to lotto combinations. Generating lotto combinations in lexicographic order is far easier than generating blackjack hands. The lotto combinations have a fixed length. For example: A lotto 6/49 game consists of 6 numbers per combination, from 1 to 49. There are 13983816 total possible elements for this type of lotto game.

Blackjack lexicographic hands can be only generated from a list, such as a deck of cards. Lotto combinations can be generated from a list as well. The list, however, must be ordered lexicographically; e.g. from 1 to 49, 49 elements, and one element per line. Blackjack hand generating face a tremendous obstacle of its own. The number of elements per combination (or per arrangement) varies widely: From two cards to 10 or 11 cards per hand.

That was the challenge that kept gambling programmers paralyzed up until now. It is so much easier to say simulation! That is, generate blackjack hands randomly! Many kids can do that, like many adults. But I reserve the right to doubt everybody's accuracy in generating blackjack hands. I've seen so much falsity out there, including pathological lying and deception. Card counting is the most blatant example. Selling card counting systems is still a sizable gambling business. All for naught!

This year of grace 2009 has proven to be augural to me. I was able to successfully finalize the software to generate all possible blackjack hands in lexicographical order. It wasn't easy. I went through various methods and algorithms. Verification is the hardest part, as there are no mathematical formulas regarding blackjack combinatorics and probabilities.

I generated blackjack hands as combinations or 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 5 years ago. The generating process is significantly faster. Also, opening large files is much easier today. My text editor of choice is my own MDIEditor And Lotto WE. It opens reasonably fast text files of several megabytes in size. The editor also uses a fixed-width font, which makes reading blackjack hands easier.

I started by creating one deck of cards. That is, write a simple text file with 52 lines, consisting of one card (number) per line, from the four 2's to the four Aces (written as 11). The file name is BjDeck1.TXT and is absolutely free to download. From that pivotal layout file (deck of cards), I deleted 4 of the 10's to get the deck used in the Double Attack Blackjack game. The new text file has only 48 entries (12 Ten-value cards, instead of 16). The file name is BjDeck2.TXT and is absolutely free to download. The output files for the arrangements generation are gigantic! Thus I needed to create smaller deck files. The reduced deck files helped me tremendously with the verification process. One deck, BjDeck1-11.TXT had only one suit of the 13 cards (one 2, one 3, ..., to 4 Tens, one Ace). The fourth deck, BjDeck4-11.TXT had only one suit of the 13 cards (one 2, one 3, ..., to 4 Tens), except for Aces. All four suites of the Ace are in that layout file (16 entries in the file).

I created two more deck layout files (also completely free to download). One file was the result of shuffling (randomizing) the regular 52-line BjDeck1.TXT: BJDeck1Shuf.TXT. The other file was the result of reversing the order in the card deck. The composition is from the four Aces to the four deuces: BjDeck1Rev.TXT. These two BJ deck files prove that the arrangements method of generating hands is the most precise. All three layout files generate the same amount of hands and the same bust percentage: BjDeck1, BjDeck1Shuf, BjDeck1Rev. By contrast, the combinations method of generating blackjack hands leads to three different result files.

1) The name of the first program is BjDealerOrder.EXE. It generates all possible blackjack hands as arrangements. That is, the order of the cards is of the essence. Total number of qualified (completed) BJ hands is staggering for all one-deck regular files: 334,490,044. As of the Double Attack Blackjack game: 302,394,480 total possible completed black jack hands. I only calculated the hands, but did not generate them. I simply commented out the statements for printing to an output file. I did generate, however, the arrangements for the BjDeck1-11.TXT and BjDeck4-11.TXT source files of incomplete decks of cards. The output files are also available absolutely freely as downloads: BjAllHands1-11Ord.TXT and BjAllHands4-11Ord.TXT.

Those two output files for incomplete card decks played a role of biblical importance, as it were. They helped me discover the most subtle errors of generating blackjack hands of variable size (length). Generating hard-count hands was as easy as a breeze. The big problem came from the Aces, as they are counted either 1 or 11. Blackjack hands such as 2+A+2+A must be hit by the dealer; hands such as 2+A+3+A are mandatory stand. I perfected my blackjack probability software due in great part to those two output files. One can easily check the error-free files in MDIEditor And Lotto WE. I have checked several times: Each and every hand is absolutely correct and no hand is missing. Of course, we follow the rules for the blackjack casino dealer. Yes, some casinos rule now that hit soft 17 (e.g. A+6) is mandatory for the dealer. First off, that's a bad rule for the blackjack player; avoid such tables at all costs. Secondly, my blackjack software can be easily adapted to accommodate the hit soft 17 rule.

2) The name of the second program is BjDealerCombos.EXE. It generates all possible blackjack hands as combinations. That is, the order of the cards is not important. Hands such as 2+A+2+A or A+A+2+2 are always written as 2+2+A+A. Total number of qualified (completed) BJ hands is far-cry lower for all one-deck regular files: 297,615 (for 52 cards). As of the Double Attack Blackjack game: 257,877 total possible completed hands. I calculated the hands and also generated them. The output files are not available absolutely freely as downloads. They require the reasonable Permanent Membership: BjAllHands1Combos.TXT and BjAllHands2Combos.TXT.

And now, the shocking results! If you go all the way down to the bottom of BjAllHands1Combos.TXT, you see that the bust percentage is 41.97%. Wow! We all believed John Scarne and his "biblical" figure of 28%! Keep this new figure in mind: The odds for a blackjack dealer's bust are at least 41.97%. The bust probability is calculated by dividing the number of dealer's busted hands to the total possible blackjack actions. Blackjack action is a parameter that counts everything: Busted hands, pat hands (17 to 21), blackjack hands, and draws or hits (incomplete hands). The software does not print the incomplete hands.

The best-case scenario regarding dealer's bust probabilities for the game of blackjack reads:

Total BJ Actions:  476246
Total Draws (Hits):178631
Total Non-Bust Hands:     97735
Total Dealer BUSTS:199880 (41.97%)
Total Complete BJ Hands:  297615
Total BlackJacks (10+A):  64 (4.83%)

The worst-case scenario is represented by the arrangements method of generating blackjack hands. The worst-case scenario regarding dealer's bust probabilities for the game of blackjack reads:

Total BJ Actions:  345948532
Total Draws (Hits):11458488
Total Non-Bust Hands:     110020276
Total Dealer BUSTS:224469768 (64.89%)
Total Complete BJ Hands:  334490044
Total BlackJacks (10+A):  128 (4.83%)

I already experienced anger and hatred from others in blackjack and gambling forums. They would fight vociferously against this very theory of mine. They would not accept these truly mathematical figures for a number of reasons. For one, the vociferous advocates are strongly involved in the business of card counting training for the game of blackjack. Some sell packages for upwards of dozens of thousands of US dollars. There are also the casinos. It is in the financially selfish interest of the casinos to advocate the application of blackjack basic strategy (BS) and, especially, card counting (BCC). The game of blackjack is highly profitable for the casinos mostly because of the BS and BCC.

And then there is that sub-human group of the afflicted. You surely know of UFO afflicts who are medically afflicted by irrationally believing in UFOs. You also know of religious afflicts who are medically afflicted by irrationally believing in gods (like "God speaks to me.") The same is true about the milder and far smaller group of card-counting-afflicted blackjack players. Some of them are medically afflicted by irrationally believing in the true effectiveness of card counting. I can only feel for such individuals. But I am truthfully harsh against them pathologically liars who use lying and deception for financial gains. Pathologically-deceiving liars who make statements on TV that they were abducted by aliens and tortured inside UFOs — and thus trying to cash in. Pathologically-deceiving liars who broadcast TV religious programs, shamelessly claiming that God speaks to them constantly — and thus trying to cash in. Pathologically-deceiving liars who shamelessly claim that they make big money by counting cards at blackjack — and thus trying to cash in. The latter three groups do it knowingly — hence the pathological-lying attribute.

There will be also probability purists who will fight the blackjack figures my new blackjack software reveals. They will argue against a number of blackjack hands generated by software: 2 2 2 2 3 3 3 or 3 3 3 2 2 2 2 or 2 2 2 2 3 3 11 11 11, or 2 2 2 2 3 11 11 11 11 3, etc. Such hands will never come out, they loudly proclaim! But, hey, who decides what hands come out and what hands will never come out? Is there a god of blackjack who makes such decisions? NOT!

• I have seen lots of search strings in the statistics of my website related to the probability to get a blackjack (natural). This time the request was personal and directed to me:

“In the game of blackjack determine the probability of dealing yourself a blackjack (ace face-card or ten) from a single deck. Show how you arrived at your answer. If you are not sure post an idea to get us started!”

• Oh, yes, I am very sure! As specified in this eBook, the blackjack hands can be viewed as combinations or arrangements (the order of the elements counts; like in horse racing trifectas).

1) Let's take first the combinations. There are 52 cards in one deck. There are 4 Aces and 16 face-cards and 10s. The blackjack (or natural) can occur only in the first 2 cards. We calculate first all combinations of 52 elements taken 2 at a time: C(52, 2) = (52 * 51) / 2 = 1326.

We combine now each of the 4 Aces with each of the 16 ten-valued cards: 4 * 16 = 64.

The probability to get a blackjack (natural): 64 / 1326 = .0483 = 4.83%.

2) Let's do now the calculations for arrangements. (The combinations are also considered boxed arrangements; i.e. the order of the elements does not count).

We calculate total arrangements for 52 cards taken 2 at a time: A(52, 2) = 52 * 51 = 2652.

In arrangements, the order of the cards is essential: King + Ace is distinct from Ace + King. Thus, total arrangements of 4 Aces and 16 ten-valued cards: 4 * 16 * 2 = 128.

The odds to get a blackjack (natural) as arrangements: 128 / 2652 = .0483 = 4.83%.

The purist argument resembles the older heated debate regarding the lotto combination 1 2 3 4 5 6. Indeed, lotto combination 1 2 3 4 5 6 does come out so rarely that it has not been drawn in our lifetime! The standard deviation plays the determinant role. The same should be true about another random phenomenon as the game of blackjack. There are some issues here that we must address. Calculating the standard deviation for lotto is as easy as it can be (just use my SUMS.EXE standard deviation software). It is extremely hard to calculate standard deviation for blackjack output files. Think of those 10 GB files! We must have a 64-bit operating system and 64-bit compilers to create adequate software. Here is the most important issue. We know exactly how to calculate the probability of any lotto combination. For example, we can generate all 13983816 lotto 6/49 combinations in lexicographic order and see exactly one 1-2-3-4-5-6 combination. We must do the same thing in blackjack: Calculate the probability precisely as p = n / N.

It took me a while to reach this moment in my activity. I am highly conscious of validity and validation. There is no doubt in my mind now that this new blackjack theory of mine is valid. I verified many times. Also, I made several verification/validation files available for free to everybody. The registered members of my website have access to additional witness files.

I do have a new blackjack strategy. I haven't applied it yet. But it should work for me even better than before. Yes, I was successful before — just keep reading. For starters, figure out the BJ dealer bust probabilities this way. 41.97% bust chance for the first hand at the table. Getting a second bust hand at the table is .4197 ^ 2 = 17.6%. How about three busted hands in the same round? .4197 ^ 3 = 7.4%. And so on ...

See how well placed the blackjack dealer is? I know, I may not place myself as well as the blackjack dealer. I can only place myself as well as the next position before the dealer. I don't know why they call it the third base ... it should be called the premier base! How many players bust before me? The more of them bust, the more ecstatic I will be to buy them a drink!

How about the software? My blackjack software to calculate accurately the odds, including the source code, was available for a limited time. The offer was withdrawn for patenting reasons — the source code was outrageously cheap.

2. The Fundamental Myth of Blackjack Gambling: Card-Counting

You might have seen that movie 21. It had absolutely no success in theaters. A DVD was released in 2008 with much more success. The 21 DVD reopened the huge gambling appetite for the so-called sure-fire strategy of counting cards at blackjack. The movie also introduced a powerfully symbolic ghost: The MIT Blackjack Team. If you watch all the features of the DVD, you will see the author of the original book that inspired the 21 movie. In his interview, the book author and the screenplay writer admitted that his book was the result of a rumor! How could people with the heads on their shoulders believe that an MIT blackjack team was even possible?! MIT (Massachusetts Institute of Technology), such a prestigious institution, would even accept the rumor of a gambling team on the premises? Let alone a real gambling team consisting of faculty and student body?! But adding MIT to a blackjack team did wonders!

The legend of card counting started with a well-written book: "Beat the Dealer!" The author, Edward O. Thorp, was a mathematician working for IBM. He also learned computer programming in order to prove his theory on blackjack card counting.

If the player keeps track of the cards that were dealt, there will be variable situations for the player. Thorp speculated that the situations were favorable to the player when ten-valued cards and Aces (high cards) were predominant in the remainder of the card deck. Reversely, the situation was unfavorable to the player when there were more small cards (2 to 6) compared to high cards. The 7, 8, and 9-valued cards were considered neutral.

The same John Scarne we talked about before puts jokingly the advantage of card counting. Suppose there is a one-deck blackjack game with 100% penetration (i.e. all cards are dealt). The player tracked the entire deck absolutely precisely. There are 5 cards remaining in the deck: 3 eights and 2 sevens. The player would bet the maximum immediately (actually, millions if it were possible!) There is NO way the player can lose! The player would always stay on two cards (it doesn't matter if it is 7+7, or 7+8) or 8+8)! On the other hand, the dealer would always bust. It doesn't matter: 7+7 (under 17); draws an 8 and busts. Or, 8+8 (still under 17); draws a 7 and busts. Or, 7+8 = 15 (under 17); either 7 or 8 as the third card would bust the dealer's hand!

A situation like that would have occurred, but extremely rarely. To have 2 sevens and 3 eights at the bottom of a 52-card deck has a degree of certainty in the same category as the moon colliding with the earth! Keep in mind, total possible permutations of 52 cards is calculated by factorial of 52 (52! = 1 x 2 x 3 x 4 x 5 … x 50 x 51 x 52). Who can say that number?!

As a matter of fact, John Scarne challenged Edward O. Thorp to a real blackjack game in a casino. I quote from Scarne's New Complete Guide to Gambling (pg. 361): “In 1964, in an effort to test Professor Thorp's 'winning Black Jack' statements I challenged him to a \$100,000 contest to be staged in Las Vegas. Thorp's reply was a big 'No'.”

This excerpt is from page 348:
"… if he [Thorp] would like to team up with me and my partner to beat the Nevada Black Jack tables by making use of his unbeatable system. Thorp agreed and after the first three days of play in Reno, Nevada, we realized that Thorp knew nothing about the science of Black Jack play, and his countdown system never seemed to work… Thorp later admitted to us that he never really gambled…”

There is one more notable name to be mentioned here: Ken Uston. He became really famous when he appeared on CBS's newsmagazine 60 Minutes. The curiosity of the TV news division was triggered by a successful lawsuit that Ken Uston had won. He charged that he was barred from playing blackjack because of his skills as a blackjack card counter. The court system decided that no player may be discriminated against based on the skills of the player. The curious thing is that Ken Uston became a spokesperson for the casino he had filed lawsuit against: Resorts. He appeared in TV commercials aired in New York and other big cities of the eastern board of the U.S. Even more curiously, the 60 Minutes Ken Uston segment was shot inside the same casino! Moreover, Ken Uston was allowed to win big money, as if were a fictional movie! Meanwhile, if a regular casino patron tries to take a benign photo on the premises, he/she might as well be thrown in jail! Not to mention that doing card counting can result in harassment and eviction by force, sometimes!

Card counting, as devised by Edward E. Thorp is a footnote to gambling history now. It offered a slight advantage in one-deck games, and especially towards the end of the deck. Ideally, a player could destroy the blackjack game if knowing the composition of the deck and the sequence of the remaining cards in the deck. The latter part is the real problem: Nobody will ever be able to know the sequence of the remaining cards in the deck. The count may be +5, but often the sequence is Low card, High card, Low, Low, High, High, Low, Low, High, etc. The dealer has the same probability to get the high cards. It is even more complicated when one considers that there are several blackjack players at the table.

The cards, Low and High, will be distributed randomly among them. What makes a particular player believe he/she would be the one to get the High cards? In one-deck games, and only playing head-to-head against the dealer, there is a slightly higher chance for the player to get a blackjack. The dealer also has an equally higher chance to get a blackjack. The difference is the player is paid 150% for a blackjack, while the dealer always gets 1 to 1 for a natural. Let's say the player has a 2% higher probability to get a blackjack (for a positive count, including Aces).

Under normal circumstances (count = 0, neutral), the probability of a blackjack is just under 5%. What is 2% of 5%? 0.02 x 0.05 = 0.001; that is, 0.1% better probability! That is, a 4.78% chance to get a blackjack, compared to 4.77%. Applying the Fundamental Formula of Gambling to p=4.77% and p=4.78%, we get, approximately, the same median: 14.18 and 14.15 hands. On the average, there will be 14 or 15 hands between blackjacks, in neutral counts, as well as in positive counts.

I have one problem with the two blackjack authors and researchers, Edward E. Thorp and Ken Uston. They reached a point of worshipping the method/system of card counting. I am sure they did a large number of computer simulations. I am sure they noticed the negligible effect of the system. Yet, they continued to influence a large number of potential casino players that card counting represented the road to the riches.

I would like to present a few more excerpts on Thorp and Uston. Carl Sifakis writes in his 1990 Encyclopedia of Gambling (pg. 36-37): “Dr. Thorp is still in computerized mathematical research, but he is now concerned with looking for values in stocks… The late Ken Uston, author of numerous books on counting, was at the end of his life involved in computer work in the Middle East, helping Kuwait track billions of dollars in investments. He was not playing blackjack in Atlantic City, although he had won a court case that barred casinos in New Jersey from refusing to let counters play. In fact, Uston, upon winning that case, didn't hit the blackjack tables in Atlantic City but signed up to do TV commercials for Resorts International, the very target of his suit… One long time gambler, Murray Friedkin, says of Thorp in 'Big Julie of Vegas' by Edward Linn: 'Thorp is the smartest man in the world; if you don't believe me, ask him…Whatever Thorp may say, I can tell you that if he has made any money on blackjack he made it by writing a book.”

Why this aura of legend surrounding card counting at blackjack? Even more mystique is added when considering that Las Vegas is still barring card counters from playing blackjack. Says Carl Sifakis: “And what of the casinos today? Blackjack is a much bigger game today than it was before 'Beat the Dealer' appeared. More people than ever patronize the tables and casinos today make more money from the game than ever before. That's a significant bottom line.” Roger Gros, senior editor of Casino Player magazine writes in his 1996 Casino Gambling the Ultimate Play-To-Win Guide: “After all, casinos make most of the money they make at the table games via blackjack. It's great advertising when someone reports a big win at the blackjack tables. Players have been encouraged for many years to believe that blackjack can be beaten, and the casinos don't want to do anything to disrupt that message.” (pg. 30.

Indeed, the legend of blackjack card counting plays as the most successful means of advertising for the casinos. It certainly attracts a large number of players who believe counting at blackjack is a road to riches. There are also other ways that the card counting legend favors the casinos. Read any blackjack book on card counting and virtually all of them contain the same cliché. “If you are a card counter, make some bonehead plays so that the pit boss won't ban you!” I think many card counters take the advice seriously. They do make bonehead, stupid plays from time to time just to hide their card-counting skills. What a stupid strategy for the blackjack player! What a profitable play for the casino!

It is admitted that counting cards offers no more than a 2% advantage for the blackjack player. That's a slim margin by all standards. Making bonehead plays can easily wipe out the slim, potential 2% advantage. The casinos owe big time to all authors of card-counting books. Then, in places where it is legal to ban skilled players from the blackjack action, the casinos commit downright robbery.

From what I have read, the casinos show a strong bias towards barring a blackjack player when he/she is at a serious loss! I read once that a known card counter was losing some \$14,000. Exactly at that point, the pit boss approached the counter and asked him to leave the blackjack game! Get it? The \$14,000 went immediately to the casino bottom line. Meanwhile, the player was deprived of a reasonable chance to recuperate his loss. After all, they say, blackjack is almost an even-odds game for a player using the basic strategy. It is fair to expect swings in the winning and losing columns of the player. Of course, the banned player is allowed to play blackjack again. The pit bosses pretend to have forgotten him/her, until another significant loss for the player. The player is thrown out again! If the known counter is winning, the rationale is that a swing in the fortune will follow. Sooner or later, the player will encounter a severe loss. That's when you ask a counter to get out! What casino would be happy if a winner takes the money and run…for real?

The 21 movie was about counting cards at blackjack. It glamorized the tactic by adding intelligence to the whole scheme: the MIT factor! The movie was shot inside the MGM Grand casino! Surprise, surprise! Would you read again the threatening message from the chairman of the same casino (2001)? He threatened my guests that the casinos would throw out casino patrons who would dare to write down roulette numbers! Yet, the same casino allowed a big movie production inside their building! A movie production worshipping card counting ...

3. The Theory of Streaks: Foundation of my Blackjack Gambling Strategy

Yes, I and casino blackjack have always had a glamorous affair. I admit, I was playing blackjack at the same tables with women who showed a strong interest in me at times. A lot of mathematics is involved and implied in this casino game. Everything that has mathematics and potential to find more about Truth attracts me as pollen attracts a hard-working bee.

I was a farm hand since my arrival to the United States in 1985. It was hard physical work. I always looked at the positive side as well. The physical work balanced my intense intellectual work, including computer programming. Unfortunately, hard physical work leads to physical pain more often than not. In early spring of 1996, unbearable back pain took me out of farm labor. I was unable to find employment until late summer of 1997.

One of the things I did was studying casino gambling. I used one of Thorp's terms: gambling mathematics. It was definitely mathematics. I saw also a means to make some money. I thought numerous times of turning into a professional gambler. There was a problem nonetheless: Bankroll. I barely had money just to survive. I am not ashamed of telling the truth. Besides, I made Nietzsche's philosophy come alive: ”What does not kill me makes me stronger.”

I did receive employment in the late summer of 1997. I worked mostly in temporary jobs. Good thing is I was paid. I was able to put food on the table and sleep under a roof. Additionally, I was able to save some money. I tried to build a bankroll for casino gambling.

Going to a casino was a freebie from where I live. The casinos offered free bus drives. Actually, you paid \$20 and the casinos gave you your money back on arrival. They are good at mathematics, aren't they? Just about everybody loses that \$20 minutes after entering the casino! Well, not me!

I hit the Atlantic City casinos a few times until the end of that year of grace 1997. I had a couple hundred dollars when I first started with blackjack. I returned home with more than the double of that amount. My window of opportunity was five hours. The bus trip would start early in the morning (Wednesday or Sunday). The trip to Atlantic City took around five hours. The trip back home would be five hours later.

The casino bosses were not mean during my first hits in 1997 - 1998. First of all, I was a small-timer: No more that 200-300 dollars per session. Besides, they might have tried to figure out my system. It was clear for them that I applied a form of Martingale betting. Most gamblers know that Martingale betting requires doubling up the bet after each loss. Well, it was a whole lot more than that and a whole lot more refined, in my case. The only negative reactions I recollect came from other gamblers.

Keep also in mind that my blackjack system was based on the wrong assumption that the dealer's bust at blackjack was 28%, as taken as a biblical figure in the gambling kingdom. As seen in another chapter, that figure is significantly higher. Therefore, the house advantage at blackjack is higher. Also importantly, the player's position at the table is relevant. At that time, I tried the middle position. By contrast, most blackjack players prefer the last position at the table (just before the dealer). That is the best position at the blackjack table: The last seat or the third base. It is a mathematical law. The more players before me bust, the lower the degree of certainty that I will bust — the BJ dealer as well. It is worth repeating. The bust probability is constant, while the degree of certainty of serial busts decreases. It is similar to consecutive heads/tails in coin tossing. Nobody gets three or four heads in a row with the same frequency as just one heads followed by one tails.

And thusly we returned to the concept of streaks once more. As I said before, probability theory might as well be named the science of streaks. The core of my casino blackjack strategy was founded on the application of streaks. The method required reasonably accurate records of the hands I played as loss or win. When I applied my gambling strategy in writing, I noted a loss by – (a minus sign) and a win by +.

Theory of probability, in general, is the science of the streaks and skips. A statement such as "This event will always have the probability equal to zero point zero zero three four etc." is virtually meaningless. The probability represents the ratio of the favorable cases over total possible cases. So, it works with integers. In real-life we deal with integers (discrete values) such as numbers of elements and numbers of trials. The events will hit, or miss, in streaks pretty clearly predicted by rules and formulas of probability theory. If you play longer sessions at the blackjack table, for example, you will face a higher probability of some very long losing streaks. But play shorter blackjack sessions, and there is a far better chance that you'll escape with shorter losing streaks. It is highly recommended to play blackjack more aggressively at the beginning of short sessions. Then, move on to another table, preferably another casino.

A session at the blackjack table can be recorded as a succession of streaks. For example, from the Player's perspective –

Win|Loss|WW|LLL|Push|L|W...etc.

The streaks can be also viewed as skips between hits; e.g. how many hands the player skipped between two hits (wins). The underpinning of the streaks is undeniably mathematical. If you don't believe me, take Warren Weaver's word on it. Please reopen his informative book, Lady Luck — the Theory of Probability at page 346. The streaks can be calculated precisely and real events concur with the formulas. Warren Weaver does the calculations by brute force. He takes the simplest and most fair game of chance: Coin tossing. Let's say we consider a total of 1024 tosses (it 2 to the power of 10; easier to work with, as the probability is 1/2). How many 4 consecutive runs of heads (H) can we toss in 1024 trials? Evidently, 4 heads should appear exactly in this pattern: THHHHT (where T stands for tails). The combined probability of that pattern is: 1/2 x {1/2 x 1/2 x 1/2 x 1/2} x 1/2 = 1/64. If we multiply 1/64 by the total number of tosses, the result is: 1/64 x 1024 = 16 runs of 4 consecutive heads (or tails, for that matter).

I wrote software to calculate the number of streaks (or runs of consecutive outcomes) of various lengths for any given probability and number of trials. The program: Streaks.exe.

My presentation here starts with the old blackjack figure of a dealer bust equal to 28%. Many blackjack players will still believe in that predicament for some time to come. I present here only the result of discarding the pushes. The approach considers the blackjack probabilities as follows:

~ Dealer: p = 52% (or 0.52);
~ Player: p = 48% (or 0.48).

Number of hands to play: 1000 (or changing blackjack tables around 5 times a day).

~ Dealer: 120 single-win streaks and 128 multi-win streaks;
~ Player: 130 single-win streaks and 118 multi-win streaks.

Noticeable differences, if playing 1000 hands continuously. My software Streaks.exe and Blackjack.EXE always enforce the rule of playing fewer hands at the same table.

Following is the total streaks for the Dealer and the Player in 1000 hands, from single streaks (1) and 2 to 8 multiple like-streaks.

Streaks	1	2	3	4	5	6	7	8	Total Wins
Dealer	120	62	32	17	9	5	2	1	505
Player	130	62	30	14	7	3	2	.8	475
The 20 extra blackjack hands are most mathematically likely allocated as 11 (52%) to the dealer and 9 (48%) to the player. Thus, the dealer can expect up to 11 consecutive wins in 1000 hands – one time; the player can expect up to 9 consecutive wins in 1000 blackjack hands - once.

The probability for dealer to bust: 28%. In a majority of cases, the blackjack dealer busts in two consecutive hands, or skipping one hand, or skipping two hands. The same is not true for the player. The basic strategy blackjack player busts 17% of the time. Therefore, the streaks of the player's bust are longer.

Let's calculate the number of streaks of losing exactly four consecutive blackjack hands (from the player's perspective). We symbolize a dealer's win by L and a player's win by W. We can calculate the number of the streaks consisting of exactly four consecutive losses for the Player.

That is, calculate the number of the streaks in the format WLLLLW. If probability for W = .48 and the probability for L = 52%, then the combined probability is: .48 * (.52 * .52 * .52 * .52) * .48 = .016845963 or 1 in 59 or 60 hands. Mathematics of gambling expects such a streak once every 60 hands, or twice in 120 blackjack hands. Or, 17 times in 1000 continuous blackjack hands. Yes, it is possible, albeit mathematically rarely, that the blackjack player can experience 3+ such streaks in 100 hands. Equivalently, however, (casino executives hate the equivalent thing!) the blackjack player could experience 100-hand sessions with all losing streaks shorter than four (3-)! Those are likely sessions with longer player's win-streaks, usually with more frequent dealer's busts!

Now, the more pessimistic reality in blackjack, from the player's perspective. I am a strong believer in the validation of any idea or theory. My latest blackjack software (2009) undeniably proves that the percentage of the bust hands is, at least, 41.97%. The bust hands were generated following the dealer's rule: Hit all hands under 16 and stand on all hands 17 or higher. In this case, the house advantage (HA) at blackjack becomes: 41.97% x 41.97% = 17.6%. That HA figure is valid only for one player against the dealer. If we deduct the traditional 4% gained by the basic strategy player, we reach a 14% house advantage at blackjack. Again, I discard the pushes (ties).

The streaks will be worse for the player under these new circumstances (which I believe to be the correct ones). No wonder so many blackjack players are dismayed to hear that blackjack is almost a 50-50 game, if playing basic strategy! The players lose far more than the so-called mathematical expectation. I did hear casino players at the blackjack tables expressing anger after quickly losing all their chips. It was better to play roulette, some said. I certainly agree with them right now. The house edge at roulette is better than in blackjack! Yes, some will always stigmatize you: "Roulette is a sucker's game!"

The 41.97%-dealer-bust approach considers the blackjack probabilities as follows:

~ Dealer: p = 57% (or 0.57);
~ Player: p = 43% (or 0.43).

The number of hands to play is the same: 1000 (or changing blackjack tables around 5 times a day).

~ Dealer: 105 single-win streaks and 139 multi-win streaks;
~ Player: 140 single-win streaks and 105 multi-win streaks.

Streaks	1	2	3	4	5	6	7	8	9	10	Total Wins
Dealer	105	60	34	20	11	6	4	2	1	1	561
Player	140	60	26	11	5	2	1	-	-	-	420
The 19 extra hands are most mathematically likely allocated as 11 (57%) to the dealer and 9 (43%) to the player. Thus, the dealer can expect up to 11 consecutive wins in 1000 hands – one time; the player can expect up to 9 consecutive wins in 1000 blackjack hands - once.

There is a significant difference that favors the blackjack dealer — that is, the casinos. A number of gamblers cautioned me with regards to John Scarne. Indeed, he helped the U.S. government with gambling matters, and especially defense against cheating the U.S. troops by gambling providers. On the other hand, he had a serious financial interest in the gambling industry. He was friends with Frank Sinatra, who, as loudly rumored, had friendship in the mob world. Perhaps John Scarne intentionally published wrong calculations pertinent to the house advantage at blackjack! The real odds would have scared away players from the most attractive and most popular casino game! Who knows?!

I was like most gamblers. I suspected dealer cheating quite a few times. Why does that dealer win so many hands in a row, while my wins come in single streaks that frequently? Believe me, I looked at dealers' hands so intensely sometimes that I risked being treated as hostile! I have never observed foul play, however. I played for fun in a Detroit casino in the summer of 2009. I spent some time with my daughter. One of my main points was the length of the streaks. Why did the dealer have so many long winning streaks (against me), while my winnings came mostly in the meager one hand a time? I remember asking my daughter if she remembered my streaks of 5 or longer hands? She was keeping track as well. I did not have a 5-streak in about 100 hands! As you see in the table above, a 5-streak of consecutive winning hands occurs 5 times in 1000 hands. That is, less than one occurrence (.5) in 100 hands.

The blackjack dealer has undoubtedly the best position at the table. No blackjack player can ever take the dealer's position. The best position for a player is just before the dealer (the last seat or the third base). The more players at the table, the better for the player in the last seat. The more players busted before the third base, the lower the bust probability for the blackjack player in the last position. The reverse is also true: No busts by the players before you, in the last position, your bust probability is higher — the blackjack dealer's, too!

If you play blackjack, be sure to sit in last chair, just before the dealer. The more players at the blackjack table, the better you are. And be sure to chip in with at least the table's maximum limit (to withstand long losing streaks). Try to play no more than 100 hands at the same table (higher probability for shorter losing streaks). You can play much more aggressively within the first 50-100 blackjack hands. If you and I happen to be at the same table, be grateful and let me sit at the third base! You can still learn a blackjack trick or two by watching me...

Hear now another new bold statement: Blackjack is a positional game of chance. The position of the players, dealer included, is of the essence. The position is next organically linked to the sequence of the cards being dealt, and therefore the blackjack hands. The degree of certainty for a busted hand depends, indirectly proportionately, to the position of the player (the lexicographic index of the sequence). That's the main reason why the blackjack dealer has the best position at the table (lexicographic sequence). One cannot apply here the probability of the binomial distribution because of the fixed direction of the card dealing. The game would be a lot fairer if the dealing of the cards would be like in holdem poker. In poker, every player will be the first, the second, … , the last hand, … , and again … , in every game. The poker strategy is different from an early hand to a late hand. The blackjack player, unfortunately, can't change the hand position or sequence… other than moving to another table…

The following passages show how I applied my streak strategy at the time when I thought the dealer bust was 28%. I also considered that player's probability was 48%, while dealer's was set to 52%. In the light of the new blackjack odds, the situation is worse for the player: 43% to 57%. In hindsight, my previous blackjack strategy was more aggressive than mathematically advised.

Yet, I was able to win consistently, up to 2003 when I reached a serious bump on the road. The floor manager of an Atlantic City casino prevented me from playing blackjack. I had been impeded to play roulette the same day, in the same casino. I did carry a pocket notebook (pencil-and-paper) and wrote down my losing and winning streaks. It was not a cheating device, by any stretch of imagination! A pencil is not an electronic device, like a miniature camera to peek at the cards in the deck! I never touched a deck of cards or a roulette wheel. I documented thoroughly the incidents and published several pages at my website: saliu.com/winning.html

You will find exact times and badge numbers and even names of the casino personnel involved. I didn't use a notebook in my previous casino outings. I memorized my streaks in this manner:

My count always starts at: 0 - 0. I say mentally zero – zero.
~ The first number represents my consecutive losses;
~ The second number represents my consecutive skips without two or more consecutive wins. This parameter is trickier to grasp. I still have difficulties keeping accurate track of it mentally. It is much easier on paper.

If I lose my first hand, the count becomes 1 / 0. I say mentally one – zero. If I lose also the next hand, the count becomes 2 / 0. I say mentally two – zero.

If I win my first hand, the count becomes 0 / 0-and-a-half. I say mentally zero – zero-and-a-half. The half simply indicates that the parameter is still running. If I lose the following hand, the count becomes 1 - 1. I say mentally one – one. That means I lost one consecutive hand; I also have a streak of one without winning two hands in a row.

If I win my first hand, the count becomes 0 - 0 and a half. I say mentally zero – zero and a half. If I win also the following hand, the count becomes 0 - 0. I say mentally zero – zero. That means I have no consecutive losses; I also won two hands in a row. Every time I win 2 consecutive blackjack hands, the second count is automatically set to 0. The mental count becomes something – zero.

If I win my first hand, the count becomes 0 / 0 and a half. I say mentally zero – zero and a half. If I lose the following hand, the count becomes 1 - 1. I say mentally one – one. That means I lost one consecutive hand; I also have a streak of one without winning two hands in a row. I lose also the following hand. The count becomes 2 & 1. I say mentally two – one.

I win the following hand. The count becomes 0 - 1 and a half. I say mentally zero – one and a half. I lose the following hand. The count becomes 1 & 2. I say mentally one – two. It means I lost one consecutive hand; I also have a streak of two without winning two hands in a row. That is, in two consecutive situations, my win was not followed immediately by another win.

Let's say I won one blackjack hand. The count is 0 - 0 and a half. I lose the following three blackjack hands. The count is 3 - 1. I say mentally three – one. I win the following hand. The count becomes 0 - 1 and a half. I say mentally zero – one and a half.

I lose the next two blackjack hands. The count is 2 - 2. I say mentally two – two. If I lose the very next hand, the left count increases by one, while the right count remains unchanged. Mentally: three – two.

I win the following hand. The count becomes 0 - 2 and a half. I say mentally zero – two and a half.I lose the next four blackjack hands. The count is 4 - 3. I say mentally four – three.

Every time I lose a blackjack hand, the left count increases by one. Every time I win a blackjack hand, followed immediately by another win, the right count is set to 0. Every time I win a blackjack hand, followed immediately by a loss, the right count increases by a half. And so on … and so forth …

With record-keeping on paper (notebook), I track the entire session. I can see the streaks of all lengths. I know I will lose 4, 5, 6, 7 … more blackjack hands in a row. I know also I will win 4, 5, 6, 7 … more blackjack hands in a row. I look at the past 30-50 blackjack hands and decide that I should expect soon a 4-5 winning streak. Or, I shouldn't expect a losing streak longer than 4-5, because I already had a longer losing streak. There is no gambler's fallacy, really; or reversed gambler's fallacy. Read the pages in the Resources section for undeniable mathematical facts.

I noticed that the second count is less streaky than the first one. That is, I win more regularly two (or more) hands in a row. I keep that in mind when I martingale my streaks. The Fundamental Formula of Gambling tells me that I win at least once in 7 hands with a degree of certainty of 99%.

I martingale more aggressively at the beginning of the blackjack game. I martingale the first count when it reaches 4. Since the second parameter is more consistent, I martingale it after it reaches 3. I go up to 7 in both cases (2-4-8 and 2-4-8-18). I stop at 7. My probability software Streaks.EXE proves the more aggressive betting at the beginning of a blackjack (or gambling) session. The degree of certainty DC for long streaks (losing or winning streaks) is lower.

I become more cautious after 100 blackjack hands or so. I martingale the first count after it reaches 6 (or 5 earlier in the game) and the second count after it reaches 5 (or 4 earlier in the game). I go as much as I can or allowed. If I sense that the streak is real bad, especially inside the same shoe, I stop the Martingale after 7, and start another one with 3 units after the previous streak ends.

The advantage at blackjack is higher payouts for blackjacks and double downs. After long losing streaks, the winnings come quite often as blackjacks (naturals) or double down hands. I also try to disturb long streaks that are favorable to the dealer. There are shoes when the dealer seems to always have blackjack, or 10-10, or never-busting hands. Some players play an extra betting box, if available. Others sit out one hand. Personally, I try not to play by the book. If playing by the book helped the dealer in this shoe, I figure I'll break the rules this hand and thus hope to break dealer's hot streak. As I said before, I split two tens against dealer's 4, 5, or 6. I also hit 12 against 4, 5, 6 in those situations. Or I stand on low count against dealer's 7-A. Unfortunately, some players get mad at me when I do that. Point is, it's my money and I play the way I want to.

My full system, as used with a notebook, keeps track of several streaks: Win (W), Loss (L), Dealer Bust (DB), Player Bust (PB), Dealer Blackjack (DJ), Player BJ (PJ). In my trimmed down version, without the notebook, I only keep track of two streaks: Win (W+), Loss (L-).

A ruled pocket notebook is the best format, especially if the number of lines is a multiple of 10 (e.g. 20 lines per page). That makes counting the winning and losing streaks much easier. I was always quick with my counts and placing my bets. I was never at fault for the flow of any game, in any casino. I did not impede any other gambler play their game.

I recreate here a fragment with my real play at Trump Plaza, July 21, 2003, beginning 5:30 PM; blackjack dealer: Ginny.

Line number	Hand parameters	Hand parameters
1, 11 	     10-	     20+
2, 12	     10+   	    50+ B
3, 13	     10-      20-
4, 14 	     10-	     25- b
5, 15	     10-	     50+
6, 16	      010-
7, 17	     30+ j    10- j
8, 18	     20-      25+
9, 19	     10+      25+
10, 20	     40+ B    10+
The interpretation.

As you can see, I did not go for the kill: My maximum bet was \$50. The minimum bet was \$10, the maximum limit at that blackjack table was \$500. The 1 to 100 ratio of the bet spread is the minimum I accept. The + sign means I won; e.g. 40+ B (line #10) means I bet and won \$40; the dealer busted that hand. The minus sign means I lost that hand; e.g. 25- b (line #14) means I bet and lost \$25, while busting. I always differentiate between a dealer bust (B) and my busted hand (b). The j means a blackjack or natural (both for dealer or player). The session was short (under 100 blackjack hands). My longest losing streak was 5 consecutive hands. My longest winning streak was 4 consecutive hands (two occurrences). I won over \$300, not counting my generous tips. I strongly advise now against tipping the casino dealers. The casinos already have the odds in their favor, not to mention the house edge (truly high at blackjack). The owners and the executives should cut their fat incomes and pay better the little guy (who takes a lot of heat to enrich the casino bosses).

Yes, things can go wrong even in short sessions. A blackjack player might encounter a really long losing streak. As a matter of fact, that's the mantra of all casino executives or gambling gurus who have a strong financial interest (selling non-mathematical systems). They will tell you that you are going to lose in very long streaks. But they never mention the opposite: The gambler can win in very long streaks. I call that the gambler's fallacy in reverse. Things happen both ways: They are mathematical, but they represent the rare exception, not the norm. If you experience a long winning streak, my advice is to leave that blackjack table as soon as the hot streak ends.

The casinos react with hostility to the new blackjack dealer's bust odds, probability calculations by Ion Saliu in 2009. Blackjack is a casino game far worse than assumed by most gamblers.

No, the casinos couldn't resist reacting ... once again! They should know by now that I don't get intimidated. Yet, they keep reacting every time I publish something of great importance. No doubt, my new research in calculating the blackjack odds of busting is ground-breaking. The new figures will surely have an impact on the game of blackjack. They even make me a criminal by association! They compare me to the criminals who become their own lawyers! Please read an hostile email I suspect originated in a casino highly paid office: Hostile Reaction by Casinos to the New Blackjack Dealer Bust Odds, Probability Calculations.

4. Blackjack Links and Resources at This Web Site

See above: The comprehensive directory of the pages and materials on the subject of blackjack, baccarat, software, systems, and basic strategy.

Most software titles require a one-time payment to download – all software for one very reasonable price. The software is then free to run forever. Pay safely online via PayPal —

Read Ion Saliu's first book in print: Probability Theory, Live!
~ Founded on valuable mathematical discoveries with a wide range of scientific applications, including probability theory applied to blackjack gambling, software, systems, new odds.

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