Why the dealer is easy to model
The dealer makes no decisions. Under the table’s rules the dealer must draw until reaching a fixed point, commonly 17, and then stop. That is a mechanical process, and mechanical processes with random inputs are what a Markov chain describes well.
A Markov chain is a set of states plus the probabilities of moving from one state to another, with one key property: the next step depends only on the current state, not on how you got there. For a dealer hand, the current state is enough to decide what happens next. Whether the dealer holds a hard 14 from a 9 and a 5 or from a 10 and a 4 makes no difference to the dealer’s drawing rule, though the exact cards left in a finite deck can differ slightly.
Setting up the states
The states are the dealer’s running hand value, with a flag for whether it is soft. A hand is soft when it contains an ace counted as 11.
- Drawing states: hard totals up to 16, and soft totals that the rules say to draw on.
- Stopping states: final totals of 17, 18, 19, 20 and 21, and a bust (over 21).
Stopping states are called absorbing: once the chain reaches one, it stays there. The task is to find the probability of ending in each one, starting from the dealer’s upcard.
Transition probabilities
Each draw moves the chain according to the card drawn. With an infinite-deck simplification the probabilities are fixed: each rank from 2 to 9 has probability 1/13, a ten-value card has probability 4/13, and an ace has probability 1/13. From a hard 12, the next card moves the dealer as follows:
- ace: to 13 (the ace counts as 1, since counting it as 11 would exceed 21)
- 2, 3 or 4: to 14, 15 or 16, so the dealer keeps drawing
- 5, 6, 7, 8 or 9: to 17, 18, 19, 20 or 21, where the dealer stops
- any ten-value card: to 22, a bust
Only the ten-value cards bust from a hard 12, with probability 4/13.
A small worked example
Take a simplified question: from a hard 12, what is the probability the dealer busts on the very next card? Only a ten-value card does it, so the answer is 4/13, about 30.8 percent. The dealer does not necessarily stop there, because the other cards keep the dealer drawing in some cases, so the full bust probability from 12 is higher once later draws are included.
The full calculation works backwards. Start with the stopping states, where the answer is known: a final 17 has probability 1 of being 17, and so on. Then compute the states just below, using the transition probabilities and the already-known results. Continue downward to the upcard. Each state’s result is a weighted average of the results of the states it can move to. This is the standard way a chain gives a complete distribution of dealer final totals.
What the result looks like
Run for a dealer upcard of 6, the infinite-deck chain under the rule that the dealer stands on all 17s gives approximate results of:
- bust: about 42 percent
- 17: about 17 percent
- 18, 19: about 11 percent each
- 20: about 10 percent
- 21: about 10 percent
These are rounded and sum to roughly 100 percent. A dealer upcard of 6 is among the weakest, a pattern that explains why some players stand on stiff hands against it. Those dealer upcard probabilities change with the rules and the number of decks.
What changes the chain
- Soft seventeen. If the dealer hits soft 17, a soft 17 becomes a drawing state instead of a stopping state, which changes every distribution slightly.
- Finite decks. In a real shoe the probabilities depend on which cards have already been dealt, so the chain needs more detail than a single running total.
- Naturals. A dealer with an ace or ten showing may hold a natural, which ends the round before any drawing. The chain is often conditioned on no natural.
Why it matters and what it does not tell you
The chain tells you how often the dealer ends in each total, and combining this with the player’s total gives the chance of winning, losing or pushing with a given decision. That is how basic strategy is justified. It describes average results over many hands, not any single hand.
Blackjack carries a house edge. Knowing the dealer’s tendencies does not remove it. Outcomes are uncertain, so play only with money you can afford to lose, and use local support services if gambling stops being enjoyable.