Blackjack basic strategy: the math behind the chart

The chart is not memorized, it is computed

Every blackjack hand is hard or soft. A hard hand has no Ace, or has an Ace that can only count as 1 without busting (for example, 10+6+Ace = hard 17). A soft hand has an Ace that still counts as 11 comfortably, like Ace+7 = soft 18, because even drawing a small card will not bust it. The difference matters because a soft hand can take a hit with zero bust risk that round, which changes the whole EV calculation. For every one of these hands, against each of the ten cards the dealer can show (2 through 9, 10/J/Q/K treated as one group, and Ace), there is a different expected value for hitting, standing, doubling, splitting (when it is a pair) and surrendering (when the table allows it). This site’s blackjack basic strategy calculator does not reproduce a chart copied from some book, it derives the chart from scratch every time you change a rule: number of decks, whether the dealer hits or stands on soft 17, whether doubling after a split is allowed, whether the table takes surrender.

The full chart covers 340 decisions: 34 reference hands (16 hard totals from 5 to 20, 8 soft totals from Ace,2 to Ace,9, and 10 possible pairs) times the 10 cards the dealer can show. Each of those 340 cells is solved separately by dynamic programming over the remaining shoe: the engine recursively simulates every possible sequence of cards the dealer still has to draw (it is forced to hit until 17), sums the probability of each final outcome (17, 18, 19, 20, 21 or bust), and uses that distribution to work out the EV of each player action. That is what separates a "strategy chart" from a list of tips: every cell answers a specific mathematical question, not a rule of thumb.

Hard hand

No Ace, or an Ace that can only count as 1 without busting. E.g., 10+6.

Soft hand

An Ace that still counts as 11 with room to spare. E.g., Ace+7 = soft 18, can hit with no bust risk.

Expected value (EV)

The average gain or loss of an action, as a fraction of the bet, if the same decision repeated infinitely.

Total-dependent

The standard convention for published charts: the dealer distribution uses the shoe minus the up-card, without removing the player’s own two cards.

House edge

The average expected loss per unit wagered, under optimal play, across a very large number of hands.

The house edge is small, but it depends on the table

House edge is the fraction of every bet the player loses, on average, playing perfect basic strategy, over a very large number of hands. It is not one single, universal number: it shifts with the number of decks, with the dealer hitting or standing on soft 17, with double-after-split (DAS) allowed or not, with surrender available or not, and above all with how a natural blackjack pays (3:2 is standard; tables that pay only 6:5 worsen the house edge by roughly 1.39 percentage points, according to Wizard of Odds). Under a liberal 6-deck rule set, dealer standing on soft 17, double after split and late surrender allowed, Wizard of Odds calculates an edge of about 0.28%. Under the typical Atlantic City rules, the same source cites 0.43%.

This site’s basic strategy calculator computes the house edge the same way, but live, for whichever exact rules you pick, instead of interpolating a generic table. Under the calculator’s own default rules (6 decks, dealer standing on soft 17, DAS, surrender and doubling on any two cards), the engine lands at about 0.40%, a bit above the 0.28% cited above because, by a simplification documented right in the code, it does not model re-splitting pairs (playing a third or fourth hand from the same pair), something more permissive tables assume is available. It is a good reminder that "the house edge of blackjack" is not a constant, it is the output of a calculation that depends on every rule at the table, including the ones that look small.

Why the dealer is a hostage to its own rules

The player chooses. The dealer does not: at every table it is forced to draw while its total is under 17 (and, at H17 tables, on soft 17 too), and forced to stop at any total of 17 or higher. There is no dealer decision, only the execution of a fixed rule. That means before the player even acts, the one piece of visible information (the up-card) already determines a full probability distribution over how the dealer’s hand will end, bust chance included. Low up-cards (2 through 6) force the dealer to keep drawing from fragile totals, so it busts often. High up-cards (7 through Ace) usually build or complete a safe total with few draws, so the dealer rarely busts.

The chart below shows the bust chance by up-card, computed for 6 decks with the dealer standing on soft 17 (this site’s calculator default rules), and it matches, within rounding, the per-up-card bust tables Wizard of Odds publishes for the same scenario. Notice the jump: from 16.7% on an Ace to over 42% on a 5 or 6, a dealer showing a 6 busts more than two and a half times as often as one showing an Ace.

That directly explains the "stand against weak, hit against strong" logic for hard totals of 12 to 16 (the so-called stiff hands). Against a 2 through 6, standing lets almost 2 out of 5 dealers bust on their own, so it is not worth risking a bust on your own hand to improve it. Against a 7 through Ace, the dealer busts less than 1 in 4 times, so standing on a weak hand is nearly always a guaranteed loss, and hitting, even with its own bust risk, has the better EV. The next section walks that math through in real numbers for two cases: the chart’s most famous one (16 against a 10) and the exact point where the call flips from hit to stand (12).

Two examples: 16 against a 10, and the flip at 12

No cell in the chart gets cited more than hard 16 against a dealer’s 10. It is the classic example because both obvious options are bad: standing on 16 loses almost every time against a dealer holding a 10 (it only needs a 7 or higher underneath to win), and hitting risks busting on any card worth 6 or more, more than half the shoe. The question is not "which action wins", it is "which action loses less", and the only way to answer it is by comparing the EV of each.

The EV of hitting is harder to write in one line because it is recursive: hitting well now can mean hitting again later. The engine resolves it like this:

1. List the possible next cardsFor every card still in the shoe, the chance of drawing it is the count remaining divided by the shoe’s total.
2. Discard the bust branchIf the new total goes past 21, that branch is worth -1 (an automatic loss): no future decision saves the hand.
3. Compare standing against hitting againIf it did not bust, compute the EV of standing on the new total and the EV of hitting once more from there (a recursive call), and keep whichever is higher.
4. Sum it all weighted by probabilityMultiply every branch by its chance of happening and add them up: the result is the EV(hit) of the original hand.

All three numbers come straight from the calculator, under its default rules (6 decks, dealer stands on soft 17). Hitting beats standing by 0.0048, less than half a cent per dollar wagered, which makes "16 against a 10" the closest call in the whole classic chart once you remove surrender from the picture. When late surrender is available, as in this calculator’s default, it beats both by a wider margin: deliberately giving up half the bet is better than risking the rest of the hand on a 16 against a strong dealer. Without surrender at the table, the correct answer is to hit, even though it feels risky: the two numbers prove that standing is slightly worse.

The second example shows the same logic flipping sides at the total of 12, one of the few hard totals that does not follow the simple "stand against 2 through 6" pattern. Against a 2 or a 3, the engine recommends hitting; against a 4, 5 or 6, it recommends standing, even though the dealer is "weak" in all three cases.

The reason is that hard 12 is too weak a hand to benefit much from a dealer that is only slightly more fragile: against a 2 or 3, the 12 itself still busts on any ten-value card (nearly a third of the shoe), so hitting still pays off even with the dealer busting "only" 35% to 37% of the time. From the 4 onward, the dealer’s bust chance has already crossed 39%, high enough that standing and leaving the problem in the dealer’s hands is worth more than risking the hand a second time.

Where this chart came from: Thorp and the IBM 704

Before a chart existed, there was only the instinct of experienced players, and it was wrong often. The first mathematically derived strategy came from Edward O. Thorp, then a mathematician in MIT’s Department of Mathematics, who in January 1961 published the paper "A Favorable Strategy for Twenty-One" in the Proceedings of the National Academy of Sciences, the journal of the United States National Academy of Sciences. A year later, Thorp expanded the idea into the book "Beat the Dealer" (1962), which became a bestseller (roughly 700,000 copies sold) and is widely cited as the moment that popularized both basic strategy and card counting.

1. January 1961The Proceedings of the National Academy of Sciences paper

   Thorp publishes the first peer-reviewed, mathematically derived blackjack strategy, showing that the game, under certain conditions, could favor the player.

2. 1962"Beat the Dealer"

   Thorp writes that "the analysis on which this book is based would have been impossible" without access to an IBM 704 computer, used to simulate thousands of hands and systematically test every decision.

3. 1966 onwardRevisions and the casino response

   The book is updated, card counting spreads, and casinos respond by tightening table rules, the same kind of rule (decks, S17/H17, DAS) that still shifts the house edge today.

It is worth separating two concepts Thorp left tangled together in popular culture. Basic strategy, the subject of this article, assumes a full, freshly shuffled shoe: the probabilities are fixed, exactly what this site’s basic strategy calculator computes. Card counting is a different thing: it tracks how the shoe’s composition changes as cards come out (more high cards left favors the player, more low cards favors the dealer), which shifts those same probabilities round by round. Counting cards is not illegal anywhere, but casinos can refuse service to anyone they suspect of doing it, so this is a historical portrait, not a tutorial. The principle of recomputing EV from the shoe’s real composition, rather than a generic one, is exactly what this site’s live blackjack advisor does with every card you enter, removing from the calculation exactly the cards already seen at the table.

From chart to table: how to read the decision blocks

Nobody memorizes 340 cells one by one. Players who run basic strategy from memory learn a handful of patterns, the same ones the math in the previous sections explains. The table below summarizes the general logic; for the exact numbers of any specific hand, use the calculator, especially because a few details break from common sense. A real example: under this calculator’s default rules, hard 11 doubles against every dealer card except the Ace, against which the correct answer is just to hit. That is the kind of exception that "memorizing the general rule" misses, and that only shows up by actually computing the EV for that specific combination.

Use these patterns to recognize the chart’s general shape, and the calculator for the exact call at whichever table you are at, with its own rules. It reacts to every rule change instantly, so you can watch visually how DAS, surrender or the deck count shift the closest-to-even cells, like the 16 against 10 and the 12 against 2 through 6 discussed above.