Player bust probability when hitting

Hitting a hard 16 busts about 61.5% of the time and hitting a hard 12 busts about 30.8% of the time, using the infinite-deck approximation in which every card rank is equally likely. Those numbers explain why stiff hands feel bad, but on their own they do not decide the play: basic strategy compares the whole expected value of hitting with the whole expected value of standing, and the dealer’s bust probability is the other half of that comparison.

Bust probability for one draw on each hard total

An infinite deck has 13 ranks, each equally likely. Ten-value cards (10, J, Q, K) are four of those ranks, so the chance of drawing a 10 is 4/13. For a hard total T, a single draw busts when the card’s value exceeds 21 − T. Counting the ranks that do so gives an exact fraction.

Two cautions on reading this table. First, it is the probability of busting on the next card only. If you hit a 12 and draw a 2, you hold 14 and may need to draw again, so the probability of eventually busting when you keep hitting to 17 is higher than the single-draw figure. Second, hard 11 and below cannot bust on one card, which is why those totals are always hit or doubled.

The table is reproduced in machine-readable form at /data/player-bust-on-hit.json, labelled as the infinite-deck approximation.

Why 12 is the dividing line

The jump from 11 to 12 is the biggest step in the game. On 11 no card busts you and a 10 makes 21. On 12 the same 10 busts you. That one-rank shift moves the bust risk from 0% to 30.8%, which is why so much of basic strategy concerns totals from 12 to 16: these “stiff” hands are the only hands in which you must choose between a real chance of busting and a hand that cannot win unless the dealer breaks.

Soft hands do not appear in the table because a soft total cannot bust on one card: the ace drops to 1 instead. Soft 18, for example, can take any card and at worst becomes hard 12 to 17. The soft hands guide explains why that freedom makes soft hands good doubling candidates.

Standing is a bet on the dealer busting

When you stand on 12 to 16 you cannot win by outscoring the dealer, because any completed dealer hand is 17 or more. You win only if the dealer busts, lose otherwise, and never push. The expected value of standing on a stiff hand, per unit staked, is therefore approximately:

EV(stand) ≈ P(dealer busts) × (+1) + P(dealer makes a hand) × (−1) = 2 × P(dealer busts) − 1.

Using the six-deck S17 dealer bust probabilities from the dealer bust table:

The Wizard of Odds six-deck S17 expected-value table gives −0.540954 for standing on 10+6 against a 10 and −0.211115 for standing on 10+2 against a 4. The simple formula lands within a few thousandths of those because the only thing it ignores is the effect of your own two cards on the dealer’s draws.

Combining the two bust probabilities

Hitting a stiff has two ways to lose: you bust yourself, or you survive and the dealer still beats you. A rough way to see the trade-off for hard 16 against a 10 in the infinite-deck model:

1. You bust immediately with probability 8/13 ≈ 0.615, losing 1 unit.
2. With probability 5/13 ≈ 0.385 you draw an ace through 5 and hold 17 to 21. Against a dealer 10, which finishes on 20 nearly 37% of the time, a 17 or 18 still usually loses, a 19 roughly breaks even, and only 20 or 21 is a clear favourite.

Working that second branch through every dealer outcome is exactly what a strategy engine does. The Wizard of Odds infinite-deck table reports the result: hitting hard 16 against a 10 has EV −0.539826 and standing has EV −0.540430. Hitting throws away 61.5% of your hands on the spot and is still the better play, because the 38.5% that survive are worth more than the 23% dealer-bust rate that standing relies on. The EV of hard 16 vs 10 article shows the numbers in money terms.

Against a 6 the sum flips. Standing on 16 has EV about −0.154, because the dealer busts 42.28% of the time. Hitting would bust you 61.5% of the time and the survivors would be facing a dealer who, when not busting, finishes on 17 more often than anything else. Standing wins the comparison comfortably, which is why basic strategy stands on every hard 12 to 16 against a 4, 5 or 6.

The 12 against 2 and 3 exception

The only stiff hands basic strategy hits against a small card are hard 12 against a 2 or 3. The bust risk is the lowest in the table (30.8%) and the dealer’s bust probability with a 2 or 3 up is the lowest of the small cards (35.35% and 37.42%). Hitting 12 against a 2 or 3 also draws a 9 for 21 or an 8 for 20, hands that beat most completed dealer hands. The combination makes hitting marginally better. The margin is thin: the six-deck S17 table gives −0.211115 for standing on 10+2 against a 4 versus −0.210364 for hitting it, a difference of less than one thousandth of a unit, which is why this hand appears in every list of composition-dependent exceptions.

Using the numbers at the table

The bust fractions are a mental model, not a rule book. The strategy chart already encodes the full comparison for the site’s profile (six decks, S17, double after split, no surrender). Where the bust fractions help is in remembering why the chart looks the way it does: the rows from 12 to 16 are the hands with real bust risk, and the columns from 2 to 6 are the upcards where the dealer’s own bust risk is high enough to make standing the better bet.

Related reading

• Dealer bust probability by upcard
• Expected value of hard 16 vs 10
• Soft hands
• Strategy chart