Blackjack basic strategy: the decisions that lower the house edge

Basic strategy is the set of hit, stand, double and split decisions that loses the least money under a given set of blackjack rules. It doesn't beat the house. Under the rules below, perfect basic strategy still leaves the house about half a percent of what you bet over the long run, and every departure from it costs more.

Where basic strategy comes from

In the early 1950s four US Army men at the Aberdeen Proving Ground in Maryland, Roger Baldwin, Wilbert Cantey, Herbert Maisel and James McDermott, set out to find the player's best play, working the numbers on desk calculators. Their paper, "The Optimum Strategy in Blackjack", appeared in the Journal of the American Statistical Association in September 1956. Its abstract notes that the strategy "differs substantially from the published strategies of card experts and the usual style of play in the casinos." A book for players, Playing Blackjack to Win, followed in 1957.

Edward Thorp, then a mathematician at MIT, started from their work and redid the calculations on a computer. His 1961 paper in the Proceedings of the National Academy of Sciences put the player's expectation for a full deck under their rules at −0.21%, against the −0.62% the desk-calculator work had produced. His bigger finding was that the odds move as cards are used up: by having the computer analyse decks with cards removed, he showed that a player who tracks what has been dealt is sometimes the favourite. His 1962 book Beat the Dealer turned that into card counting.

The rules this chart is for

Basic strategy depends on the rules, so these are the ones assumed here. They are the rules of Blackjack on cs2.casino:

• An infinite deck: every card is drawn independently, so each rank has a 1 in 13 chance and a ten-value card (10, J, Q or K) a 4 in 13 chance.
• The dealer stands on all 17s, soft 17 included.
• Blackjack pays 3:2.
• You may double on any first two cards, including the first two cards of a split hand.
• You may split a pair once, and split aces get one card each.
• With an ace or a ten showing, the dealer checks for blackjack first, so a dealer blackjack takes only your original bet.
• Insurance is offered against a dealer ace. There is no surrender.

I worked the chart out exactly for these rules by comparing the expected value of every option, then checked it against the Wizard of Odds' tables for infinite-deck blackjack: all 600 of its hit, stand and double figures match to the six decimal places it prints. It is nearly the same as the Wizard's chart for four decks. The two differences are soft doubles: with an infinite deck, A,2 against a 5 and A,4 against a 4 are better hit than doubled.

The chart

H means hit, S stand, D double (hit if you can't), Ds double (stand if you can't) and P split. Columns are the dealer's up card. With three or more cards, use the row for your total.

Since you can split only once, a new pair dealt to a split hand is played as a hard total.

The decisions people get wrong

Some of these feel wrong at the table:

• Hit 12 against a 2 or 3. Standing on 12 against a 2 gives up about 4% of your bet compared with hitting.
• Hit 16 against a 10, though it's the closest call in the chart: the two options differ by 0.06% of your bet.
• Hit soft 18 against a 9, 10 or ace. A total of 18 isn't safe against those cards.
• Hit 11 against an ace rather than doubling. Under these rules hitting returns about 3.4% of your bet more.
• Always split aces and 8s, and never split 5s or 10s.

Why insurance is a poor bet

Insurance is a side bet of half your stake that pays 2:1 if the dealer's hidden card is a ten. With an infinite deck the chance of that is exactly 4/13, whatever else is on the table. For each coin you insure, the expected result is 2 × 4/13 − 9/13 = −1/13, a loss of about 7.7%. That is a far bigger edge than the main game carries.

Taking "even money" when you hold a blackjack against an ace is the same bet in another form. Insured, you're paid 1 unit whatever happens. Uninsured, you get 1.5 units with probability 9/13 and a push otherwise, worth about 1.038 units on average, so declining is better by roughly 3.8% of your bet.

Why counting can't work here

Counting works because a real shoe changes as it's dealt: as cards are used up, the odds on the next hands shift, sometimes in the player's favour. That was Thorp's insight. With an infinite deck nothing is used up. Every card is a fresh, independent draw, so the chance that the next card is a ten is 4/13 no matter what came before, and there is nothing to count. The same independence means streaks don't change the next hand either, as the gambler's fallacy explains.

What the edge means for you

For these exact rules my calculation gives a house edge of 0.57% with perfect basic strategy, measured against your initial bet. Two Wizard of Odds figures support it: his house edge calculator gives 0.49% for the same rules with eight decks, its largest option, and his infinite-deck tables give 0.51% with resplitting allowed. Allowing only one split costs roughly another 0.06%. Call it about half a percent; the game page shows the house edge it uses.

In money, 1,000 hands at 1 coin each lose about 5.7 coins on average, and over a short session luck swamps that figure in either direction. The edge shows itself over thousands of hands, as house edge and RTP explains, and no betting pattern changes it, as betting systems shows. The chart is still worth learning, because mistakes cost far more than the edge. To practise, cs2.casino accepts a zero bet, which stakes nothing.

Sources

• Expected Returns in Infinite-Deck Blackjack — Wizard of Odds
• Blackjack House Edge Calculator — Wizard of Odds
• Blackjack Strategy for 4 Decks — Wizard of Odds
• The Optimum Strategy in Blackjack — Journal of the American Statistical Association
• A Favorable Strategy for Twenty-One (1961) — eScholarship, UC Irvine
• Four Horsemen of Aberdeen — Wikipedia
• Edward O. Thorp — Wikipedia