The basic strategy is derived by taking the factors that are known (the value of the player’s hand and the dealer’s upcard) and applying probability to determine the likelihood of the unknown factors (the value of the dealer’s hole card and the value of each “hit” card). Since there are 13 different cards, and an equal number of cards of a given value, there is a 1 in 13 chance that any unknown card will be of any of the values. When the likely values of unknown cards (any hit, plus the dealer’s hole card) are added to the values of cards that are known (the player’s initial hand and the dealer’s up card), the probable outcome on both sides of the table makes the correct decision self-evident.

Likely Values of the Casino Dealer’s Hand

The value of a dealer’s hand is the total of his upcard plus the value of the hole card. While the value of the hole card is unknown, it has an equal chance of being any of the thirteen possible values, which gives some indication of the total values of the dealer’s hand. For example, when the dealer’s upcard is a nine, here are the possible hands:

HOLECARD 2 3 4 5 6 7 8 9 10 J Q K A
VALUE 11 12 13 14 15 16 17 18 19 19 19 19 20

In this instance, 7 out of the 13 possible values (53.85% of all possible hands) have a value of 17 or higher. If the player stands on any total under 17, he will lose more than half the time the moment the dealer’s hole card is overturned. If the player hits to 17, only 6 of 13 of these hands will defeat him—meaning that he will win more than half the time.

This is part of the reason that basic strategy tells the player to hit any hand, hard or soft, against the dealer’s nine until he has reached a total of 17 or higher—but it is only one factor to be considered.

Likely Outcomes of the Casino Dealer’s Hand

Since the casino dealer will hit any hand until it reaches a certain total (in these examples, a hard 17), it is not enough to consider the initial value of the dealer’s hand, but the value it is likely to reach after all hits are taken.

For example, if the dealer’s upcard is a three, the range of possible initial values is 5 to 14, with a 7.39% chance of having each possible value each except 13, which is four times as likely (30.77%) because there are four ten-value cards. In all of those cases, the dealer will take a second hit because the value of the hand has not reached 17. Form thirteen possible outcomes when the hole card is overturned, there are now 169 possible outcomes after the first hit:

HIT 2 3 4 5 6 7 8 9 10 J Q K A
On 5 7 8 9 10 11 12 13 14 15 15 15 15 S16
On 6 8 9 10 11 12 13 14 15 16 16 16 16 17
On 7 9 10 11 12 13 14 15 16 17 17 17 17 18
On 8 10 11 12 13 14 15 16 17 18 18 18 18 19
On 9 11 12 13 14 15 16 17 18 19 19 19 19 20
On 10 12 13 14 15 16 17 18 19 20 20 20 20 21
On 11 13 14 15 16 17 18 19 20 21 21 21 21 12
On 12 14 15 16 17 18 19 20 21 22 22 22 22 13
On 13 15 16 17 18 19 20 21 22 23 23 23 23 14
On 13 15 16 17 18 19 20 21 22 23 23 23 23 14
On 13 15 16 17 18 19 20 21 22 23 23 23 23 14
On 13 15 16 17 18 19 20 21 22 23 23 23 23 14
On S14 S16 17 18 19 20 21 12 13 14 14 14 14 S15

After the hit is taken, there are 169 possible outcomes.

  • In 24 (14.20%), the dealer will bust
  • In 65 (38.46%), the dealer will stand on a hand of 17-21
  • In 80 (47.34%), the dealers’ hand will remain below 17

From there, the possible outcomes of the 80 hands that still need to be hit will need to be calculated, weighted, and added to the total. Some of those hands will remain under the required total and will require another hit, and so on.

The dealer will be compelled to take up to 9 hits in a multi-deck game (3 + 2 = 5 then 2, 2, 2, A, A, A, A, A, A). To determine the odds of the outcome of the dealer’s hand, the likelihood of each value resulting from each of those hits, times the likelihood that the hit would need to be taken, must be added to the original total.

This requires a daunting amount of math, but it can be done with the help of a computer to arrive at an aggregate conclusion. Given an upcard of three, considering all likely outcomes of all likely hits, here is the likelihood of all possible outcomes:

17 18 19 20 21 BUST
13.51% 12.99% 12.50% 12.44% 10.92% 37.64%

Likely Outcomes of the Player’s Hand

The likely outcomes of a player’s hand must also be computed. The method for doing this is exactly the same as the one just illustrated for computing the likely outcomes of a dealer’s hand, except that the value of both cards in the hand is known at the onset (So the first step can be elided).

It’s still necessary to derive the possible outcomes after all hits to know if hitting is in your advantage. If you’re dealt a hard 13 (three-ten), the likelihood of possible outcomes is:

17 18 19 20 21 BUST
9.61% 9.61% 9.61% 9.61% 9.61% 51.96%

Deciding to Hit or Stand

In deciding whether to hit or stand on a given total, the likely outcomes of both the player’s and dealer’s hands must be compared:

17 18 19 20 21 BUST
PLAYER: 9.61% 9.61% 9.61% 9.61% 9.61% 51.96%
DEALER: 13.51% 12.99% 12.50% 12.44% 10.92% 37.64%

If the player were to stand, he would only win when the dealer busts, so his chances of winning are 37.64% (there is no chance of a tie).

To calculate the player’s chances of winning if he were to hit the hand, consider all possible of all possible outcomes in which …

  • the dealer busts and the player does not (18.08%)
  • the dealer draws to 17 and the player draws higher (5.19%)
  • the dealer draws to 18 and the player draws higher (3.74%)
  • the dealer draws to 19 and the player draws higher (2.40%)
  • the dealer draws to 20 and the player draws higher (1.20%)

When totaled, the player who hits a hard 13 against a dealer’s three has a 30.61% to beat the dealer’s hand.

When comparing the two possibilities, it is clear that the player has a better chance of winning by allowing the dealer to bust (37.64%) than he does by hitting his hand (30.61%). Therefore, basic strategy charts indicate that, when holding a hand of thirteen against a dealer’s three, the player should stand.

Deciding to Surrender

In some cases, neither outcome (hit or stand) is favourable to the player, and he stands to lose less money in the long run if he simply quits the hand.

This would seem to be the case in the previous hand—even if the player follows the best possible strategy, he will lose 62.36% of all hands in which he is dealt a 13 against a dealer’s three.

However, when a player surrenders, he only reclaims half his wager. The player who surrenders a 13 against a three over the course of 100 rounds of play will lose 50 betting units. If the player were to see the hand through, he would lose 62.36 betting units, but win 30.64 betting units (no loss/gain in the 5.47% of hands that tie), for a net loss of only 31.72 betting units. Clearly, the net outcome is better for the player who plays the hand through rather than surrendering.

The break-even point for playing versus surrendering is exactly 25%. At those odds, the player who follows through on the hand in 100 rounds of play will win 25 units and lose 75, for a net loss of 50 units—the same as if he surrendered the hand. The odds of winning dip below 25% in very few instances: for example, when a player is holding 15 or a 16 against dealer’s ten. In these cases, surrender is the best option.

Deciding to Double

In deciding whether the player should double his wager, the odds of losing with a single hit must be compared to the odds of winning when taking as many hits as necessary to make a hand.

For example, a player who opts to hit a hand of 10 versus a dealer’s seven stands a 57.84% chance of winning, a 32.09% chance of losing, and a 10.07% chance of tying the dealer’s hand. Over the course of 100 rounds of play, the player will win 25.75 betting units.

On the first hit alone, the player stands a 56.08% chance of winning (he cannot bust a 10, and he has a 7.69% chance for a 17, 7.69% for 18, 7.69% for 19, 30.77% for 20, 7.69% for 21), a 36.40% chance of losing, and a 7.52% of tying. Over the course of 100 rounds of play, the player will win only 21.48% of all hands dealt—but since the wager is doubled, this amounts to 42.96 betting units.

In order to maximize his profits over the course of the game, the player should double a hand of 10 against a dealer’s seven.

The effects of doubling strategically are dramatic. The fact that basic blackjack strategy instructs a player to stand on a low total in situations where the dealer is likely to bust trims the house advantage, but what really closes the gap is doubling in situations where it is advantageous to do so.

Deciding to Split

When the player is dealt a paired hand, he has the option of playing it as it lays (a pair of sevens has an initial value of 14) or creating two separate hands, the values of which are not known at the time the player must make the decision.

As when assessing the dealer’s probable outcomes, it’s necessary to first determine the likelihood of the second card in each of the player’s hands, then the likely outcome of all possible hits, both compared to the dealer’s possible outcome based on his upcard.

For this example, let’s say the dealer’s upcard is a seven. If the player keeps his own sevens together, he plays the hand as a 14. His likely outcomes are: lose 54.42%, win 38.49%, tie 7.09%. Over the course of 100 rounds of play, the player who does not split sevens will lose 15.93 betting units. (This is another unfortunate situation, but not to the degree that surrender is warranted.)

If the player splits his sevens, the likely value of the resulting hands can vary from 9 to 18 (two through ace as the second card, with four tens making a 17 four times more likely). The aggregate chances (comparing all possible hands to all possible hits) yield a 40.68% chance of losing, a 42.05% chance of winning, and a 17.27% chance of tying the dealer.

Not only have the odds turned in the player’s favor, but 100 rounds of play will result in 200 individual hands because one hand is split into two separate and independent ones, each with a 1.37% advantage. As a result, the player stands to win 2.74 betting units by splitting sevens against the dealer’s seven over the course of 100 rounds of play.

Developing the Charts

The basic strategy charts are developed by

  1. Deriving every possible outcome for the player on 26 hands (17 hard, 9 soft) plus the 13 instances in which a player may receive a paired hand.
  2. Deriving every possible outcome for the dealer, based on the 13 possible upcard values.
  3. Considering the impact of every possible decision (hit, stand, surrender, double, or split) in every possible situation.
  4. Recommending the best possible course of action.

As these examples have borne out: sometimes the “best possible course of action” will enable a player to lose less in a bad situation; other times, it will enable the player win more in a good situation; and on rare occasions, it can even turn a losing situation into a winning one.