Probability - the MATHEMATICS OF CHANCE.


The concept of probability is fairly easy to grasp. Probability is an indicator or a numerical measure of the LIKELIHOOD that an event will occur. Thus, probabilities could be used as measures of the degree of uncertainty associated with events associated with the game of blackjack for which, as a player, you need to make certain decisions that would make you win, such as:

  • What are the “chances” of getting a 10 or an Ace for my “double down”?
  • What is the “likelihood” of getting a favorable card when hitting my hard 15?
  • How “likely” is it that the dealer’s hole card is a 10 so that buying insurance would be a good option?
  • What are the “odds” in favor of raising my bet to the maximum during the next round?

Probability values are always assigned on a scale from 0 to 1. If the event cannot possibly happen, we say that the probability is equal to zero. If the event is sure or certain to happen, then the probability is equal to 1.  A probability near 0 indicates that the event is UNLIKELY to occur; a probability  near 1 indicates that an event is VERY LIKELY or ALMOST CERTAIN to occur. Other probabilities between 0 and  1 indicates  varying degrees of likelihood that an event will occur. The following figure illustrates this view of probability:

Probability as a Numerical Measure of Likelihood of an Event

Probability can be expressed as a fraction, a concept most everybody is familiar with:

Probability (Event) = P (Event) = N/D

where the numerator, N, represents the ways the event can occur SUCCESSFULLY, and the denominator, D, represents the total number of ways the event can occur both SUCCESSFULLY and UNSUCCESSFULLY. As a fraction, probability  can also be expressed as a decimal. Thus, in Tossing or flipping a coin, we will observe one of two possible results: get a HEAD or TAIL. With a fair coin,  getting a HEAD or getting  a TAIL are equally likely. Hence, the probability of the event “get a HEAD”, is given by

P(HEAD) =  1/2 = 0.50

Experiments and the Sample Space

In discussing probability, we first define a (PROBABILITY) EXPERIMENT as any process that results in well-defined outcomes. When the experiment is repeated, one and only one of the possible experimental outcomes can occur. Several examples of such experiments in the gaming world are found in the following table:


In analyzing a particular experiment, it is necessary to carefully  identify the experimental outcomes. The set of all possible experimental outcomes is usually referred to as the SAMPLE SPACE for the experiment; any one of the experimental outcomes is called a SAMPLE POINT and is an element  of the sample space.

It should be mentioned that each outcome of a probability experiment occurs at random.  This means that you cannot predict with certainty which outcome will occur when the experiment is conducted. In addition, each outcome of the experiment is equally likely to occur. This also means that each outcome has the same probability of occurring.

It is also important to point out that the notion  of a probability experiment is somewhat different from the “experiments” conducted in science laboratories. In the laboratory, the researcher assumes that each time an experiment is repeated in exactly the same way, the same outcome will occur. When we speak of probability experiments, the outcome is DETERMINED BY CHANCE, such that even though the experiment might be repeated in exactly the same way, a different outcome may occur.

For this reason, probability experiments may also be referred to as RANDOM EXPERIMENTS.


In discussing probabilities, it is usual to consider several outcomes of the experiment.  In our earlier example of rolling  a single die, we may want to consider getting an odd number -  a 1, 3 or 5. We  refer to this as the event of getting an odd number  from the experiment of rolling a single die. Thus, an event  would consist of one or more outcomes of the sample space.   An event with only one outcome is called a simple event. An event with  two or more events is called a compound event. 

In the experiment of drawing a card from a standard single-deck, we can list the outcomes of  several events, as shown in the following  table:

Assigning Probabilities to Experimental Outcomes

In assigning probabilities to experimental outcomes, two basic requirements / axioms must be satisfied:

  1. The probability values assigned to each experimental outcome (or sample point) must be between 0 and 1.  Denoting by Ei the ith experimental outcome and P(Ei) its corresponding probability, we must have

0 ≤  P(Ei)  ≤ 1  for all  i

  1. The sum of all of the probabilities must be equal to one. That is,

P(E1) + P(E2) +  . . . P(Ek)  = 1

Any method  that satisfies the above requirements and results in reasonable numerical measures of the likelihood of the outcome is acceptable.

In practice, the classical or objective method, the relative frequency method, or the subjective method are often used. This will be discussed in our next post.



Yes! You Can Win at Blackjack

Welcome to How-to-Win-at-Blackjack.org!

What the Ed Thorpes and Ken Ustons of yesteryears (some 40 A natural blackjack hand.to 50 years ago) have clearly proven is that, indeed, you can "Beat the Dealer" and Win at Blackjack!

But, blackjack today is no longer the same, you say? That is true, we have to admit.

And, ironically, Thorpe and Uston, and all the other premier members of The Blackjack Hall of Fame - Julian Braun, Al Francesco, Peter Griffin, Tommy Hyland, Lawrence Revere, Max Rubin, Arnold Snyder, Keith Taft, Edward O. Thorpe, Ken Uston, Stanford Wong - successful writers and professionals who have heralded their exploits to the world  in winning at blackjack, have primarily themselves to blame for the drastic and substantial changes instituted by the Casinos to counter the innovations - primarily Card Counting and Team Play - introduced by those Blackjack Greats to beat blackjack.

Though hand-held single-deck and double-deck blackjack games are still around today, single-deck games, dealt to the bottom of the deck, are gone. Taking over are the multi-deck (4,6,8 decks) blackjack games dealt from "shoes" with  the cut card set at 50% -75% penetration, games with a thousand variations - Over/Under 13, Royal Match, Super Sevens, Lucky Ladies, Spanish 21, Super Fun 21, Double Exposure, Blackjack Switch, and  cards shuffled by machines (and worse, by those infernal continuous shuffling machines).

Though blackjack, like all other casino gambling games, is set up so that the House always has an edge over us players - WE MAINTAIN THAT YOU CAN STILL WIN AT BLACKJACK. 

Even today! For several reasons which have not changed – the characteristics being inherent in the game of blackjack itself.

Unlike other games, blackjack is a game composed of many variables that can affect exactly how much of an edge the House has.

For one thing, blackjack is unique in that the house edge depends completely on how you, as a player, play your hands. You may choose to hit, stand, double-down, split pairs, take insurance, and in some casinos, surrender your hand, giving up half of the amount of your bet.

Since a non-thinking and self-destructive player could theoretically play at a 100% disadvantage by hitting all hands, including blackjacks, until he busts, the leading practitioners in blackjack developed what is now well-known as Basic Strategy – a  complete set of decision rules covering all possible choices that the player may encounter, based solely on the dealer's up card and without consideration to other player's cards.   Basic Strategy lowers the house edge (arising from how a player plays his hands) to a minimum 0.5% (which means that in the long run, you lose only 50 cents for every $100 wagered.)

State-of-the-art research in blackjack in recent years, however, shows that by taking into account information on other players' cards (such as the dealer's hole card and the other players' face down cards of today's single-deck and double-deck games – yes, it is possible to determine what the other players’ face down cards are without actually exposing them!) to modify certain play actions in Basic Strategy makes a huge difference, not only turning around the edge in favor of the player but do so – considerably!

Another variable that affects the house edge is that blackjack is the only casino game in which the odds continually change back and forth between the player and the house. The reason for this is that blackjack does not follow what is referred to by statisticians as the "Law of Independent Trials."

Let us explain. In the game of Roulette, each spin of the wheel is independent of the other spins made on the wheel. In the same way, one particular throw of a fair pair of dice in the game of Craps has absolutely nothing to do with what you get in a succeeding throw. Unlike the casino games just mentioned, in blackjack what has happened in the past affects what happens in the future. For example, when you draw a "10" to your hand, the deck of cards now has one less "10" left to play. This means that one particular card that has been removed from the deck will certainly affect what happens to your hand in your subsequent play, and much more so when a number of such "10"cards have been similarly removed.

Blackjack researchers have shown that when relatively more tens and Aces have been played, the remaining deck favors the house. In contrast, when many more small cards relative to aces and tens have been played, the deck favors the players. To be sure, there are several other significant variables affecting the player's odds that come into play as current on-going research continues to reveal. Incorporating these new information into our mode of play can only increase our chances in winning at blackjack.

Having a way to track what cards have been removed from the deck provides you another way of increasing your edge over the Casino which you need to win at blackjack. You can leverage the knowledge on what cards have been removed to your advantage. With the odds in your favor, you can then increase your bet and at the same time vary your play based on the information that the deck is not only high cards-rich but other information on the other cards that affect your likelihood to win the round.  Similarly, when the deck favors the house, you can place just the minimum bet, and also correspondingly vary your play.

The resulting edge for the player over the house has been shown to range from 0.5% to 7%.

One major roadblock on the Road to Winning Blackjack, however, is the way traditional card counting has been and is being implemented. While the use of multi-level card-counting systems (which assign a number of different point values to various cards) have been shown to provide substantial THEORETICAL advantage for the blackjack player, implementing them in a casino environment is no piece of cake – adding or subtracting several and different values to a running count, which only a few guys like Ken Uston or the other Blackjack Hall of Fame greats and perhaps a very small percentage of the blackjack community can do.

NO WONDER, then, why card-counting practitioners have fallen back to the simpler one-level, balanced count, systems such as HI-LO (which assigns +1 to Low Cards, -1 to High Cards, and 0 to what are believed as neutral cards – the 7s, 8s, and 9s –  computing a running count by adding or subtracting 1 in increments, and subsequently converting it to a true count by using an estimate of how many cards are left in the deck) or the unbalanced systems,  KO (Knock-Out) and Red Sevens - with one modification of assigning  a  “+1” to the 7s in KO and “1/2“to the red 7s in Red Sevebs, but with no need to convert a running count into a true count to be useful.

But while seemingly simple enough for a number of players, these methods are NOT FOR AVERAGE GUYS, like the rest of us,  who can count only in the way we were taught to  count:  "by 1’s,” that is, . . ., 1, 2, 3, 4, . . etc., or "by 10s”, that is, . . .10, 20, 30, 40, . . etc.,  and to make simple one-digit  or two digit additions or subtractions (12 + 5, 9 +8, 15 + 7, 21 – 14, 21 – 9, etc.,. . .)  Facility with the simple additions just indicated enables us to enjoy part of the thrill in playing blackjack -  that is, KNOWING INSTANTLY when we get a hit card that our hand is GOOD and to a lesser  degree . . . has BUSTED.

THE BIG QUESTION THEN IS: Is Winning Blackjack beyond the average guys?  OF COURSE NOT!

THE REASON WHY NOT:  The purpose of card counting is to determine whether or not a greater proportion of  LOW CARDS have been dealt in the earlier rounds,  leaving the HIGH CARDS in greater numbers in the remaining cards to be dealt – thus giving us the EDGE during the next round of play. To do this, OUR METHOD merely COUNTS the LOW CARDS as they are dealt (NEVER MIND THE HIGH CARDS – they will still be there and we will know how many there are) and count them the OLD-FASHIONED WAY USING COUNTING NUMBERS,  . . . 1, 2, 3, 4, 5, . . . YES, . . .the way you have been taught to count . . . and NOT . . ., 0, +1, +2, +1, 0, +1, 0, - 1, -2, . . . etc. . . . up and down, the old, traditional card-counting way.

AND, to provide you with INSIGHT WHY AND HOW OUR METHOD WORKS, we will look at Blackjack (just like the other casino games - craps, roulette, baccarat, poker, etc. ) as a GAME OF CHANCE, and SHOW and TEACH you the MATHEMATICS involved -  the PROBABILITIES, ODDS, and EXPECTED VALUES, etc. –  No, we will not use the high-powered mathematics  of Nobel Prize winners like John Nash (with A Beautiful Mind) but MATHEMATICS which YOU and YOUR 5TH GRADER can discuss, argue, and agree about.

One comment: It is not too early to point out at this stage that the computation of the probability value of drawing a “favorable card” as your next hit card, such as a 7 for a hard hand of 14,  or a group of cards, such as the HIGH CARDS for a “double down” on your hard 11,  requires information on the exact number of that card or of the HIGH CARDS, as the case maybe, remaining in the deck of undealt cards, the probability being equal to a simple ratio between the high cards left and the number of cards remaining. This is why OUR METHOD requires you to actually count EITHER the LOW CARDS or the HIGH CARDS (with the low cards being our choice), NOT BOTH. Looking at or counting both HIGH and LOW CARDS as used in HI-LO or Knock-Out  is a tremendous and unnecessary burden to average or ordinary guys, leading a significant number of them to flee from or quit the game  altogether. Knowing the number of cards dealt so far (and this can also easily be determined), we can actually compute everything you need to know on the remaining deck to guide your betting and playing decisions.

A Natural Reaction to the Above Comment: Your immediate reaction might be, “What? Do you expect me to be computing probabilities during the game? Nuts! There are over a thousand situations that you would normally encounter in Blackjack. This is worse than the simple up and down additions in HI-LO or K-O!”

Our Answer: Our probability-based method involves defining what you call a PIVOT NUMBER for any given number of cards dealt, a number which when reached by your actual count of the LOW CARDS will show that the state of the cards being dealt in which the probability of getting low cards in the remaining deck is equal to 50% and over is passing to the complementary state in which the probability of getting high cards would now be equal to or over 50%. The computation is an inherent part of the counting process and you actually do not compute probabilities, except in highly critical or exceptional situations – which as we have claimed before your 5th grader can readily do.

One last comment: With your new understanding of BASIC PROBABILITY CONCEPTS, you will see DEFINITIVE PROOF that the basic assumption of considering the 7s, 8s, and 9s (“the middle cards”) as NEUTRAL CARDS and therefore have been and are IGNORED by traditional card-counting methods (and for that matter, any card similarly not taken into account in the card-counting process), IS FAULTY as it may LEAD TO RISKY PLAYING DECISIONS by users of traditional card-counting methods.

Indeed, the house edge can be overcome and you can win at blackjack, but not without DISCIPLINE, KNOWLEDGE, PRACTICE & PREPARATION, and the MENTAL SET to put into play your pre-determined betting and playing strategies.

Elements of Winning Blackjack

The requirements to winning at blackjack include:

  • Basic Strategy - The dealer's up card provides the initial information you need to make a reliable decision about what you should do with your hand - to hit, stand, double down, split or surrender as the casino rules allow!
  • Card Tracking and Analysis - a way of gaining an edge over the house by tracking the cards dealt and using the "value" of the cards remaining to be dealt during a game to indicate how you should modify your basic strategy play and to determine how much you should be betting on your next hand. As implied in our discussion above, we shall be using a probability-based counting procedure, using simple counting . . . 1, 2, 3, 4, . . ., etc.  of the low cards that have been dealt. [It is somewhat like Speed Count of Golden Touch Blackjack, which is based on the HI-LO system with its inherent flaw and consequences – meaning the middle cards as being neutral and ignoring them altogether by assigning them with the value of zero – but MUCH, MUCH MORE!]
  • Smart Money Management - being smart about how you handle your money: knowing how much to bet, understanding your limits (both emotionally and monetarily), how to protect your bankroll and staying within your comfort zone on both fronts.
  • Best Practices of the Experts / Pros - part of a general plan that they follow when gambling, regardless of the game, which you should also keep in mind for your own play.

Your mastery and corresponding execution of the essential elements to win at blackjack -  namely, basic strategy, card tracking and analysis, smart money management and best practices, whether IN PERSON or IN TEAM PLAY, can mean not only the difference  between your WINNING and LOSING, but between SIMPLY WINNING and WINNING BIG at Blackjack!

Look for our various posts for more details on each of these elements to guide you on your quest for how to win at blackjack at the casinos and online. And, thank you for visiting with us.


Dr. Pete, Your Host

Dr. Pete,  Your Host.


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