Web site by Ion Saliu, founder of lottery software science, gambling mathematics.

Primer: Blackjack Basic Strategy, Mathematics, Card Counting

By Ion Saliu, Blackjackologist At-Large

The best of blackjack is at blackjack theory site.

Published on September 3, 2002.

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. The cards two to 9 have a numerical value equal to the number printed on the card. Tens also have a numerical value equal to the number printed on the card (10, of course). All face cards (Jack, Queen and King) have the value of 10. Aces (A) may be counted as either 11 or 1.

The rules of blackjack are relatively simple to learn. This page is not meant, however, to be a blackjack tutorial A dealer plays against a number of players, from one to seven. Every player and the dealer receive initially two cards each, dealt by the dealer. Each player's hand is played against the dealer's hand only. If a player's hand has a value closer to 21 (without going over) than the dealer's hand, the player wins. The best possible hand is known as a blackjack (twenty-one in the first two cards). This hand consists of an A (ace) and a ten-valued card (10, J, Q, K). The payout for a blackjack is 3-to-2: the player is paid three chips for every two chips bet. When both the player and the dealer have blackjacks, it is a normal tie (push) situation; the player retains the initial bet.

The player has several choices after receiving the first two cards.
1) Hit or draw: take one or more cards to add-up to a better hand
2) Stand: stop taking more cards
3) Double down: double the initial amount (in cases considered more favorable)
4) Split pairs: if the two cards are equal in value they may be played in two separate hands

The dealer has limited choices. The dealer must draw until dealer's hand adds-up to 17 or more. The dealer is not allowed to stand on 16 or less. The dealer is not allowed to draw to a hand summing up to 17 or more.

It is also important to know how to hand signal player's actions.
- Hit: tapping the table next to your cards with your finger tips or by placing your hand near your cards and moving your finger tips toward you (somehow like a gesture of grabbing)
- Stand: moving your hand left to right, in a flat, palm down position about two inches over your cards (like waving off)
- Split: place another bet, equal to your initial bet, alongside your initial bet; do not touch your cards, the dealer will split your cards
- Double down: you ante up another bet equal to and alongside your first bet; the dealer will deal you one more card (only one), across the first two cards.

Both the player and the dealer can go over 21. The situation is known as bust. The player loses the bet immediately. The dealer plays his/her hand last, after all the players at the table. This rule creates the so called house advantage or house edge or percentage advantage. John Scarne was the first to calculate the house advantage at blackjack: 5.9%. There are several bonehead plays, however. Such foolish plays can increase the house advantage to 10%-15%. Example of such play: hitting a hand of 19 or 20!

The house edge can be diminished to only around 1% if the player follows certain rules. The set of rules known as basic strategy make blackjack one of the fairest games of any kind, almost as fair as coin tossing. I strongly recommend that you do not attempt to play blackjack before you learn basic strategy.

An IBM computer scientist, Edward O. Thorp, published “Beat the Dealer” in 1962. The book introduced a winning method known as card counting. The method considered the ten-valued cards and the Aces as “positive”, and the cards 2 to 6 as “negative”. If the net result in the remaining deck was positive, the player must increase the bet accordingly. The method had visible results when only one deck was used and very few cards remained in the deck. The casinos modified the blackjack rules dramatically. The “penetration” was introduced: not all the cards in the deck are played. Shuffling is done unexpectedly. Also, most casinos introduced the multiple deck blackjack. Card counting is now very similar to cult believing, like religious fundamentalism or UFO believing. I can only recommend you read my message

  • The myth of blackjack: card counting; real math. You can also use my blackjack software for more testing. Then, make your own reasonable decision.

    My recommendation is to learn basic strategy and the Fundamental Formula of Gambling (FFG). I created several pages to help you learn basic strategy. Color-coding is the key. Learning blackjack basic from spreadsheet-like tables is much harder. Only a few blackjack players have mastered the basic strategy. The rest can learn a lot easier if follow color-coded charts first. Then, they can look up tables for reinforcing and perfecting. I compiled all the color-coded charts and the basic strategy table in one page. Then I created one page for each color-coded basic strategy and one page for the summarizing strategy table. The links are located at the end of this primer. The color-coded charts are created around the main actions a blackjack player can take. Thus, there is a split pairs chart; there is a double down chart; and there is a color-coded hit or stand chart. The action is colored red. For example, if the recommended action is split pairs, the corresponding situation is colored in the highly visible and memorable red colour. Your first two cards are shown in the left column. The dealer's up-card (face card) is shown in the top row. Let's say, your first hands are 7, 4. The dealer face card is 6. Look first at the split pairs chart; it does not apply, you don't have a pair. Look next at the double down chart. Your cards add up to 11. There is an entry in the left column. The intersection (cell) of row 11 and column 6 is colored red. The recommended action is therefore double down.

    You will notice that dealer's card 6 represents an important demarcation (frontier) line. Most player's actions are dramatically different for dealer's cards 2 through 6 compared to dealer's cards 7 through A. That's the key of learning basic strategy. The color red of the actions makes learning even easier.

    I do not present any of my blackjack strategies here. Keep in mind that the cards have streaks as any other gambling phenomenon analyzed by FFG. The decks have a high degree of clumpness at the beginning of the game. It is difficult to shuffle the new card decks to a high degree of randomness in a minute or so. Therefore long streaks of like-cards come out after the initial shuffle. The clumpness diminishes after several shuffles. Try to get a seat in the first position (next to dealer's left hand side). If you track the streaks of cards (how many highs and how many lows) you can make changes to the basic strategy. Try to get a seat in the last position (next to dealer's right hand side). That way you can see everybody's cards. Perhaps a high card is more likely; stand on 16 against dealer's 10, even if the basic strategy requires a hit (draw). You know you'll bust anyway, why not get an extra chance…

    Finally, my recommendations. Software: Masque Blackjack. Books: New Complete Guide to Gambling by John Scarne; Best Blackjack by Frank Scoblete.

    • Ion Saliu's Paradox and Gambling

    Ion Saliu's Paradox of N Trials is presented in detail at saliu.com, especially the probability theory page and the mathematics of gambling formula. If p = 1 / N, we can discover an interesting relation between the degree of certainty DC and the number of trials N. The degree of certainty has a limit, when N tends to infinity. That limit is 1 - 1/e , or approximately 0.632…

    If you play 1 roulette number for the next 38 spins, common belief was that you expected to win once. Not! Non! Only if you play 38 numbers in 1 spin, your chance to hit the winning number is 100%. Here is an interesting table, which includes also 'The Free Roulette System #1" in the last row.

    Type of Play Winning Chance % Cost in Units Gain for Saliu's Paradox %
    Play 1 # in 38 Spins 63.7% 38 -
    Play 38 #s in  1 Spin 100% 38 1 - .637 = 36.3%
    Play 34 #s in  1 Spin 89.5% 34 28.8%

    The maximum gain comes when playing 38 numbers in one spin: 36.3%. Obviously, it makes no sense to play that way because of the house advantage. On the other hand, a so-called wise gambler is more than happy to play one number at a time. What he does is simply losing slowly! Not only that, but losing slowly is accompanied by losing more. That cautious type of gambling is like a placebo. A roulette system such as 'Free #1' scares most roulette gamblers. "Play 34 or 33 numbers in one shot? I'll have a heart attack!" In reality, the 'Free System #1' offers a 28.8% advantage over playing singular numbers in long sessions. In fact, the gain is higher since the bets are placed at optimal moments. That's mathematics, and there is no heart to worry about.

    You can also use SuperFormula to calculate all kinds of probabilities and advantage percentages. The option 'L — At least M successes in N trials' is a very useful instrument. If you play 19 numbers in one spin, the probability to win is 50%. If you play 19 numbers in 2 consecutive spins, the probability to win 'at least once' is 75%.

    Advantages of Ion Saliu's Paradox in the game of blackjack.

    Gambling math, card counting, blackjack basic strategy.

    Blackjack: Software, Systems, Basic Strategy, Card Counting

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

    Web site by Ion Saliu, founder of lottery software science, gambling mathematics.


    Web site by Ion Saliu, founder of lottery software science, gambling mathematics.

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    Web site by Ion Saliu, founder of lottery software science, gambling mathematics.