MathBench

Blackjack Odds Calculator

See the dealer's exact bust and outcome odds for every upcard, and the expected value of standing, hitting, or doubling any hand — computed from the game's actual probabilities, not folklore.

Dealer outcome probabilities

How this calculator works

Dealer mode shows the full outcome table: for each upcard, the exact probability the dealer finishes on 17 through 21, turns over a blackjack, or busts. This is the table basic strategy is built on — the dealer's forced, choice-free algorithm (draw to 17, stand) makes their fate pure probability, and the recursion behind this page just follows every branch of it.

Hand mode answers the live question: given your total and the dealer's upcard, what is each option really worth? It computes the expected value per dollar of standing, of hitting (followed by optimal play afterward), and of doubling — and names the best. The famous surprises are all in there: hard 16 versus 10 is nearly a coin-toss between two bad options, and "always stand on 12 against a weak card" turns out to be true against a 4 but false against a 3.

The formula

Dealer: draw ranks with P = 1/13 (tens 4/13), hit until ≥ 17
        → outcome distribution by exact recursion

EV(stand, T) = P(dealer bust) + P(dealer < T) − P(dealer > T)
EV(hit, T)   = Σ P(card) × [ bust → −1, else max(EV stand, EV hit) ]
EV(double)   = Σ P(card) × 2 × EV(stand after that card)

The infinite-deck model — every draw from a fresh shoe — is the standard analytical simplification in the founding texts (Edward Thorp, Beat the Dealer; Peter Griffin, The Theory of Blackjack); its EVs differ from an 8-deck shoe by hundredths of a percent. This page's engine reproduces the published infinite-deck values exactly — hard 16 vs 10 returns −0.5404 standing and −0.5398 hitting, the canonical photo finish.

Worked example

The classic: hard 16 against a dealer 10, S17 (the hand-mode default):

  1. Dealer's no-blackjack distribution showing 10: busts 23.0%, finishes 20 or 21 41%+
  2. EV of standing: −0.5404 — you win only when the dealer busts
  3. EV of hitting: −0.5398 — you usually bust, but occasionally improve to 17–21
  4. Difference: 0.0006 per dollar — six hundredths of a cent
  5. Verdict: hit, by the narrowest margin in the game

Both options lose about 54¢ per dollar over the long run — 16 vs 10 is simply a bad place to be. The calculator's honest contribution is showing that "the book says hit" is a photo finish, not a law of physics — and that surrender, where offered, beats both.

Assumptions & tips

  • Rules move the odds more than luck rituals do. 6:5 blackjack payouts cost you about 1.4% — triple the entire house edge of a good 3:2 game. Read the table placard before sitting down.
  • The dealer's 4–6 are your doubling window. Bust rates north of 39% are why basic strategy doubles and splits aggressively there — the chart is this page's table in disguise.
  • Never take insurance (see the FAQ) — and "even money" on your blackjack is insurance with better marketing.
  • Expected values are long-run averages. A −0.54 EV hand can still win tonight; the math describes thousands of hands, not the next one. Bet accordingly, with money whose loss is entertainment expense.
  • Perfect play ≠ profit. Basic strategy shrinks the house edge to roughly half a percent; it never crosses zero. The calculator exists to make the game transparent, not beatable.

Frequently asked questions

Where does the house edge in blackjack actually come from?

From order of play: you act first, and if you bust, you lose immediately — even when the dealer busts right after. That double-bust asymmetry is worth roughly 6 to 8 percent to the house before your options claw it back. Blackjack paying 3:2, doubling, and splitting recover most of it, landing the game near a half-percent edge with perfect basic strategy — and several times that with casual play.

Why should I never take insurance?

Insurance is a side bet that the dealer's hole card is a ten, paying 2:1. With an ace up, the dealer has a ten underneath 4/13 of the time — 30.8 percent — but a 2:1 payout needs 33.3 percent to break even. The gap makes insurance a roughly 7.7 percent house-edge wager regardless of your hand. "Even money" on your own blackjack is the same bet wearing a disguise.

Is using this calculator card counting?

No. These are the fixed probabilities of the game before any cards are seen — the math behind the basic strategy card that casinos happily sell in their own gift shops. Card counting means adjusting bets as the deck composition changes during play; it is legal but unwelcome. This page teaches the baseline arithmetic every player is entitled to know.

What does the "infinite deck" assumption change?

It treats every draw as coming from a full shoe, ignoring card-removal effects. For a 6- or 8-deck game the differences are a few hundredths of a percent — far smaller than the effect of rule variations like H17 versus S17 or 6:5 blackjack payouts. Every number here is exact under the model; the model itself is the standard textbook simplification used in Thorp's and Griffin's analyses.

Why doesn't the calculator cover splitting pairs?

Split EVs depend on what happens across two (or more) new hands, including resplits — a genuinely bigger computation whose answers are already distilled into any basic strategy chart. The stand/hit/double comparison here covers the decisions where seeing the actual EVs teaches the most, like discovering that hard 16 versus 10 is a 0.0006 photo finish.

Sources

  1. The Optimum Strategy in Blackjack — Roger R. Baldwin, Wilbert E. Cantey, Herbert Maisel and James P. McDermott, Journal of the American Statistical Association, volume 51, number 275, pages 429–439, 1956. DOI 10.1080/01621459.1956.10501334. The first published derivation of basic strategy from the dealer's fixed drawing rule, and of the stand / hit / double comparison this calculator recomputes. Cited bibliographically: the journal copy is paywalled.
  2. A Favorable Strategy for Twenty-One — Edward O. Thorp, Proceedings of the National Academy of Sciences, volume 47, number 1, pages 110–112, 1961. pmc.ncbi.nlm.nih.govThorp's computer search for the player's best possible strategy, dispensing with the desk-calculator approximations of Baldwin et al. — the peer-reviewed origin of the exact expected-value approach used in hand mode. Its complete-deck expectation of −0.21% (against Baldwin et al.'s −0.62%) is the source of the tips' point that fixed-strategy play stays below zero; Thorp's own contribution was showing that tracking the cards already dealt is what turns it positive.
  3. Beat the Dealer: A Winning Strategy for the Game of Twenty-One — Edward O. Thorp, revised edition, Vintage Books, 1966. Named in this page's formula section as a source of the infinite-deck simplification that both the dealer recursion and the player EV recursion assume. Cited bibliographically: no free full text to link.
  4. The Theory of Blackjack: The Compleat Card Counter's Guide to the Casino Game of 21 — Peter A. Griffin, 6th edition, Huntington Press, 1999. ISBN 978-0-929712-13-0. The standard analytic treatment of blackjack expected values and card-removal effects, cited in the formula section for the published infinite-deck figures this engine reproduces and for the size of the error the infinite-deck assumption introduces. Cited bibliographically: no free full text to link.
  5. Exact Calculation of Expected Values for Splitting Pairs in Blackjack — John A. Nairn, arXiv preprint arXiv:1909.13710, 2019. arxiv.orgBacks the FAQ's answer on why pair splitting is left out: resplit expected values need a separate and far heavier recursion than stand, hit and double, and exact complete results for splitting arrived only recently.

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