Dealer bust odds by upcard, on an infinite shoe

The shoe with no memory

A physical blackjack shoe is a finite thing: six or eight decks, and every card that comes out changes the mix of what is left. That is what makes card counting possible and what makes exact odds a moving target. The engine here does something simpler and, for the purposes of a table like this one, cleaner. Each card is derived from its own cursor of the round’s hash as floor(u × 52), so every draw is an independent pick from a full deck. Ranks two through nine and the ace each have probability 1 in 13; a ten-value card has 4 in 13.

With those two facts, independent draws and stand on all 17s, the dealer’s hand is a small Markov chain that can be solved exactly. Start from the upcard, draw until the total is 17 or more, counting a soft total as an ace plus ten while that keeps the hand under 22. The figures below are computed for this article by that recursion, not simulated and not copied from a strategy site.

The table

The 21 column for a ten or an ace includes the dealer’s natural. Rows sum to 100 within rounding. Weighted by how often each upcard appears, the dealer busts on about 28.2% of hands.

Read down the bust column and the game’s structure appears. From a 2 the dealer must draw at least twice to reach 17 and busts about a third of the time; each higher low card makes the second draw more dangerous, and a 6 is the worst card the dealer can show, busting 42.3% of the time and, when it survives, most often stopping on a weak 17. Then the cliff: a 7 needs only a ten to make 17, and the dealer stands on 17 more than a third of the time from it. From a ten, the dealer makes 20 on a third of hands. From an ace, the dealer busts once in nine and makes 21 more than a third of the time.

Read across a row and you see why a player’s stiff hand is a different problem against different upcards. Against a 6, standing on 16 wins whenever the dealer busts, 42% of the time, and the dealer’s made hands are mostly 17s. Against a 10, standing on 16 wins only the 21% of the time the dealer busts, and the dealer’s made hands are mostly 20s. That asymmetry is the whole content of the top half of a basic strategy chart.

The player’s side

Each row is the number of busting ranks over 13, with tens counted four times. A hard 16 is the worst hand in the game for exactly this reason: hitting it busts more often than not, and standing on it beats only a dealer bust.

Put the two tables together and the classic decisions fall out without any memorising. Hard 16 against a dealer 6: hitting busts 61.5% of the time, standing wins the 42.3% of the time the dealer busts, so stand. Hard 16 against a dealer 10: standing wins only 21.2% of the time, and hitting, though it busts more often than not, at least sometimes produces a hand that beats a 20, so hit. Hard 12 against a 4: hitting busts only 30.8% of the time while the dealer busts 39.4%, and the right answer is close enough that charts differ on the neighbouring cells.

• The tables are exact for this engine and approximate for a real shoe. A finite shoe with cards removed shifts every figure slightly, which is the effect a counter tracks. Here there is no removal.
• The dealer’s peek matters for one thing only. Because the dealer checks for a natural under a ten or an ace before you act, a dealer blackjack ends the hand before a double or split can lose extra; the bust odds above are for hands that continue.
• None of this beats the edge. Basic strategy on these rules returns about 99.5%, the figure the engine header states; the tables explain the strategy, they do not improve on it.