Why tables are built rather than guessed
Every published figure for blackjack, from the percentage of hands that bust to the cost of a rule change, comes from one of two sources: exact calculation or simulation. Both start from the same idea. Write down the rules precisely, then work out what happens on average when those rules are followed many times.
Blackjack is awkward to solve by hand because the cards are drawn without replacement and the player makes decisions along the way. A computer can handle this by brute force, either by enumerating every possible sequence of cards or by sampling a very large number of them at random. The second approach is simulation.
What a simulated round looks like
A simulation program contains a virtual shoe, a set of rules and a playing strategy. For each round it does roughly the following:
- Shuffles the shoe, or draws from the cards that remain.
- Deals two cards to the player and the dealer.
- Applies the player’s strategy: hit, stand, double, split or surrender.
- Plays out the dealer’s hand according to the dealer drawing rules.
- Records the result as a number of units won or lost, such as +1, -1, 0 or +1.5 for a natural paying 3 to 2.
Repeat this millions of times, add up the results and divide by the number of rounds. The average result per round is the estimate of the player’s expected return, and its negative is the estimate of the house edge for that rule set and strategy.
From raw results to a table
A strategy chart is produced by running the same machinery many times with different choices. To decide whether to hit or stand on a hard 16 against a ten, the program plays the situation repeatedly with each option and compares the average outcomes. Whichever option loses less on average goes in the chart. Doing this for every combination of player hand and dealer upcard fills out the grid.
The same approach gives dealer upcard probabilities, such as how often a dealer starting with a six ends up busting. Under common assumptions that figure is a little over 40 percent. The exact number depends on the rules and the number of decks, which is one reason published tables rarely agree to the last decimal place.
Randomness and the size of the sample
A simulation uses a pseudo-random number generator to stand in for shuffling. Each run is a random sample, so the answer wobbles a little from run to run. The wobble shrinks as the number of rounds grows, but slowly.
Blackjack results per round have a standard deviation of a little over one unit, while the house edge is only a fraction of a percent of a unit. To estimate the edge to within about 0.001 units (0.1 percent of a one-unit bet) with reasonable confidence, you need roughly a million rounds. To narrow it ten times further you need about a hundred times more rounds. That square-root behaviour is why serious tables are built from very large runs and why small differences between two published numbers may just be sampling noise.
Simulation versus exact calculation
Where possible, authors prefer exact methods. The dealer’s final total, for instance, can be calculated exactly from the cards remaining, because the dealer has no decisions. Player decisions with complicated rules, such as resplitting several times with limits, are harder to enumerate and are often handled by simulation or by a hybrid of the two. Exact results have no sampling error; simulations have a measurable one.
What goes in matters as much as the sample size
A flawless simulation of the wrong game gives a precise but useless answer. Before trusting any odds table, check the assumptions:
- the number of decks and whether the dealer stands or hits on soft seventeen
- the payout for a natural
- doubling, splitting and surrender rules
- whether the dealer checks for a natural before the player acts
- the strategy used, since a table assumes the player follows it perfectly
- whether cards are dealt from a shoe with a cut card or from an endlessly reshuffled supply
Changing any one of these changes the result. That is why the same game can appear with slightly different edges in different sources, and why reading odds tables carefully means looking for the stated assumptions rather than just the headline figure.
What a table cannot tell you
An odds table describes an average over a very long run. It says nothing certain about the next hand, the next hour or the next trip. Short-term results swing widely around the average because of statistical variance, and a player can lose over any short stretch regardless of how well the strategy is followed. Past hands also do not change future odds, so a table cannot reveal a “due” outcome.
Blackjack has a house edge under standard rules, even with perfect basic strategy. Simulation can measure that edge and show which decisions cost less, but it cannot turn a negative expectation into a positive one. Gambling can cause real harm, so play only with money you can afford to lose, and if gambling is becoming a problem, contact a local support service.
Reading simulation claims sensibly
When you meet a figure that claims to come from simulation, ask a few questions. How many rounds were played? Were the rules stated in full? Was the strategy fixed or varied? Is the figure given with a range, or only as a single number? Honest sources usually answer these, and a result presented with no rules and no sample size deserves caution.