Blackjack Odds and House Edge: A Rules-Based Derivation

A blackjack table’s cost to the player is not a single number. It is a function of the rules printed on the felt, the number of decks in the shoe, and how the player responds to the cards. The same game that returns 99.5% under one rule set returns 98.0% under another, and the difference is not hidden in a side bet or a shuffle. It is in the payout for a natural and in the dealer’s obligation on a soft 17.

The figure that matters is the house edge: the expected loss per unit wagered. It is not a property of the cards. It is a property of the payout table and the strategy applied to it. This page derives that edge from the outcomes, states every rule the figure assumes, and converts the percentage into money at a stated stake and round rate.

The outcomes and their payouts

A standard blackjack hand has three terminal results for the player: win, lose, or push. The payout for a win is not uniform. A natural blackjack pays 3:2 at most tables, but 6:5 at many tables that advertise a low minimum. A regular win pays 1:1. A push returns the stake. A loss takes the stake.

The house edge comes from two sources: the dealer acts last, so the player who busts loses before the dealer draws, and the player must act first on incomplete information. The dealer’s fixed strategy of hitting to 17 and standing on 17 or more is not a choice. It is a rule. The player’s choice is whether to follow basic strategy, which is the set of decisions that maximises expected value against a given rule set.

Deriving the edge from the payout table

The edge cannot be derived from the payout table alone, because the probability of each outcome depends on the composition of the deck and the strategy used. But the contribution of the blackjack payout can be isolated.

Consider a single-deck game where the dealer stands on soft 17 and blackjack pays 3:2. The probability of being dealt a natural blackjack is:

(4/52) × (16/51) × 2 = 0.0483, or 4.83%.

The factor of 2 accounts for the two orders: ace then ten, or ten then ace. The probability the dealer also has a natural is approximately 0.0483 × (15/50) × (3/49) × 2, but for the purpose of the payout contribution, the relevant comparison is the difference between 3:2 and 6:5.

At 3:2, a 1 unit bet returns 1.5 units profit on a natural. At 6:5, it returns 1.2 units. The difference is 0.3 units per natural. Multiply by the probability of a natural:

0.3 × 0.0483 = 0.0145, or 1.45% of the initial bet.

That 1.45% is added to the house edge when the payout drops from 3:2 to 6:5. The exact figure varies slightly with deck count and composition, but the magnitude is consistent: 6:5 blackjack costs the player roughly 1.4% more per hand than 3:2, before any other rule is considered.

For the full house edge, the calculation requires enumerating every hand and the optimal decision at each point. The standard figures, derived from combinatorial analysis, are:

These figures assume the player follows basic strategy exactly. They also assume no surrender, no double after split, and no resplitting aces. Each of those rules changes the edge by a few hundredths of a percent, and the direction is always in the player’s favour when the rule is added.

Converting the edge to money per hour

A percentage of turnover is not a quantity most players can feel. The same 0.5% edge costs a different amount depending on how many hands are played and how much is wagered per hand.

Assume a player bets 10 units per hand, plays 60 hands per hour, and faces a 0.5% edge.

Step 1: Total turnover per hour = 10 × 60 = 600 units.
Step 2: Expected loss per hour = 600 × 0.005 = 3 units.

At a 10 unit bet, that is 3 units per hour. If the unit is 1, the expected loss is 3.00 per hour. If the unit is 25, the expected loss is 75.00 per hour.

Now apply the same calculation to a 6:5 table with a 2.0% edge.

Step 1: Turnover = 10 × 60 = 600 units.
Step 2: Expected loss = 600 × 0.02 = 12 units per hour.

The difference between the two tables is 9 units per hour at the same stake and round rate. Over a three-hour session, the 6:5 table costs 36 units more in expectation. That is the cost of the rule, expressed in money.

The number of hands per hour matters as much as the edge. A full table with seven players and a slow dealer might deal 40 hands per hour. A heads-up game with a fast dealer might deal 200. At 200 hands per hour and a 0.5% edge, the expected loss at 10 per hand is 200 × 10 × 0.005 = 10.00 per hour. At 40 hands per hour, it is 2.00 per hour. The edge is identical. The cost is not.

What people get wrong about blackjack odds

The most common error is to treat the house edge as a fixed property of the game rather than a property of the rule set. A player who has read that blackjack has a 0.5% edge may sit at a 6:5 table and believe the figure still applies. It does not. The 6:5 payout alone adds about 1.4%, and the dealer hitting soft 17 adds another 0.2%. The table that looks like blackjack is not the game the figure describes.

This mistake is natural because the game is presented under one name. The felt says blackjack. The cards are the same. The dealer’s motions are the same. The only difference is a line of text in the corner of the table, and that line is often printed in small type. The player who does not know to look for it has no reason to suspect the game has changed.

The second error is to believe that a previous run of results changes the next hand. A shoe that has produced ten consecutive player wins does not owe the player a loss, and a dealer who has busted five times in a row is not more likely to bust again. Each hand is dealt from a depleted deck, and the composition of that deck is the only thing that matters. The cards have no memory. The shoe is not due to produce anything. The probability of the next hand is determined by the cards remaining, not by the cards already played.

This mistake is natural because the human mind is pattern-seeking. A sequence of wins looks like a streak, and a streak feels like it must end. But the deck does not know the streak occurred. The only thing that changes the odds is the removal of specific cards, and that effect is small until the deck is deeply depleted.

The third error is to believe that a staking system can overcome the edge. A Martingale progression doubles the bet after a loss, which changes the distribution of outcomes: many small wins, occasional large losses. The expected value is unchanged. The house edge applies to every bet, and no sequence of bet sizes alters the mean. A player who doubles after a loss is not improving the return. They are increasing the variance and the size of the eventual loss.

This mistake is natural because the progression produces a high frequency of winning sessions. The player who doubles after a loss will win most sessions, because a single win recovers all previous losses plus one unit. The rare session where the progression meets a long losing run wipes out many small wins. The arithmetic is not hidden. It is just distributed unevenly across time.

Card counting as history, not method

Card counting is the practice of tracking the ratio of high cards to low cards in the remaining deck. A deck rich in tens and aces favours the player, because blackjack pays more than 1:1 and because the dealer is more likely to bust. A deck rich in low cards favours the dealer.

The method was developed in the 1960s and became widely known after the publication of Beat the Dealer. It works in the sense that a player who bets more when the count is favourable and less when it is unfavourable can shift the expected value positive. The shift is small, typically 0.5% to 1.5% depending on the count system and the rule set.

Casinos responded by shuffling earlier, using more decks, and banning known counters. A six-deck shoe with a penetration of 75% gives the counter fewer opportunities than a single-deck game dealt to the last card. The modern game is designed to reduce the effect of counting, not to eliminate it entirely.

For the purposes of this page, counting is history. It explains why shoes are shuffled when they are, why some tables use continuous shuffling machines, and why the rule set at a low-minimum table is often worse than the rule set at a high-minimum table. It is not a method recommended here, and it is not a method that changes the house edge for a player who does not count.

The rules that determine the figure

Every house edge figure on this page depends on the following conditions:

• The number of decks in the shoe. More decks slightly favour the dealer.
• Whether the dealer stands or hits on soft 17. Hitting soft 17 adds about 0.2% to the house edge.
• Whether blackjack pays 3:2 or 6:5. 6:5 adds about 1.4% to the house edge.
• Whether the player may double after split. Allowing it reduces the edge by about 0.1%.
• Whether the player may surrender. Late surrender reduces the edge by about 0.07%.
• Whether the player may resplit aces. Allowing it reduces the edge by about 0.08%.
• The strategy the player uses. Basic strategy is assumed. Any deviation increases the edge.

A figure stated without these conditions is not a figure. It is a number that happens to be associated with a game called blackjack. The same game, played under different rules, costs the player different amounts per hour, and the difference is large enough to matter.

Worked example: the cost of a 6:5 table

A player sits at a six-deck table where the dealer stands on soft 17 and blackjack pays 6:5. The player bets 20 per hand and plays 80 hands per hour. The player follows basic strategy.

The house edge for this rule set is approximately 1.94%. The expected loss per hour is:

Step 1: Turnover = 20 × 80 = 1,600.
Step 2: Expected loss = 1,600 × 0.0194 = 31.04 per hour.

Now compare the same player at a six-deck table where the dealer stands on soft 17 and blackjack pays 3:2. The house edge is approximately 0.54%. The expected loss per hour is:

Step 1: Turnover = 20 × 80 = 1,600.
Step 2: Expected loss = 1,600 × 0.0054 = 8.64 per hour.

The difference is 22.40 per hour. Over a four-hour session, the 6:5 table costs 89.60 more in expectation. That is the cost of the payout rule, expressed in money at a stated stake and round rate.

The player who does not know the rule exists cannot choose between the two tables. The player who does know can read the felt before sitting down. The arithmetic does not require an operator name or a recommendation. It requires the rule set and a calculator.