Blackjack math appendix

Start with the core formulas, because every practical number in this course flows from three or four foundational expressions.

Blackjack mathematics is not exotic. It uses expected value, variance, and a handful of derived measures that any player willing to spend an afternoon with a spreadsheet can verify. The academic lineage runs from the 1956 Baldwin paper (the first correct strategy calculation) through Edward Thorp's 1962 computer simulations, to Peter Griffin's The Theory of Blackjack (1979), Stanford Wong's Professional Blackjack (1975, multiple editions), and Don Schlesinger's Blackjack Attack: Playing the Pros' Way (2005 third edition). These are the primary sources. The Wikipedia article on card counting is a reasonable introductory summary but should not be cited as a primary source for precise figures.

Expected Value, Edge, and House Advantage

Expected value per hand is the weighted average outcome across all possible results. For a basic strategy player against a six-deck S17 game at 0.50% house edge, the expected value of a £100 bet is £100 x (1 - 0.0050) = £99.50 returned on average. Equivalently, the expected loss per hand is £0.50. Over 500 hands, the expected total loss is £250. This is the baseline calculation that applies to all edge discussions in this course.

The house edge changes with count. Each unit of true count above neutral adds approximately 0.50% to the player's edge in a Hi-Lo framework. This figure, sometimes called the "value of a unit of true count" or the "count advantage per true count," is derived from simulation studies. The most commonly cited figure is 0.50% per true count unit, though the precise value varies slightly with rule set and penetration. Our analysis covers both the Hi-Lo system structure and the approximate edge-per-count-unit figure.

Effective edge over all hands is the figure that matters for bankroll and session calculations. This is the overall edge averaged across every hand played, including the large fraction played at minimum bet during neutral or negative counts. A counter with a 2% edge at TC +4 who bets 12 units in those conditions, and a 0% edge at TC +1 who bets 2 units, and a -0.50% edge at neutral count who bets 1 unit, has an effective edge computed by weighting each edge by the probability of that count occurring and the bet size placed. The effective edge in a typical six-deck UK game with a 1-to-12 spread is approximately 0.50-0.75% for a well-calibrated counter with good penetration.

Standard Deviation, Variance, and N0

The standard deviation of a single blackjack hand is approximately 1.15 units when splits, doubles, and blackjacks are accounted for. The variance (standard deviation squared) is approximately 1.32. These are theoretical figures based on full basic strategy play; actual variance in a counted game with a spread is slightly higher because the bigger bets in high counts add more variance per session.

For a session of N hands, the standard deviation of total outcome is: SD(session) = 1.15 x sqrt(N) units. For 200 hands at a £25 unit, the standard deviation of outcome is 1.15 x sqrt(200) x £25 = approximately £407. Your expected loss at 0.50% house edge over 200 hands is £25 x 200 x 0.0050 = £25. The standard deviation dwarfs the expected value at this scale. This is why individual session results are nearly meaningless as a signal.

N0 (N-zero) is defined as: N0 = variance / (expected value per hand)^2. For a counter with an effective edge of EV per hand and variance V per hand: N0 = V / EV^2. At EV = 0.0075 (0.75% edge) and V = 1.32: N0 = 1.32 / (0.0075)^2 = 1.32 / 0.0000563 = approximately 23,440 hands. N0 is the horizon at which the expected profit equals one standard deviation of results; it represents the approximate point at which the edge begins to statistically dominate variance in the cumulative result. The derivation and application of N0 is developed most fully in Schlesinger's Blackjack Attack.

Risk of Ruin Formulas

The continuous risk of ruin formula for a player with a positive edge is derived from the Gambler's Ruin problem in probability theory. The standard form is:

RoR = e^(-2 x EV x B / V)

where EV is the expected value per hand as a fraction of the unit, B is the bankroll in units, and V is the variance per hand.

At EV = 0.0075 (0.75% edge per unit), V = 1.32, and B = 200 units: RoR = e^(-2 x 0.0075 x 200 / 1.32) = e^(-2.273) = approximately 0.103, or about 10.3% risk of ruin. To get below 5% at those edge and variance parameters, you need approximately 260 units. To get below 1%, you need approximately 440 units. our risk of ruin calculator confirms these figures.

This formula assumes fixed bet sizing (one unit throughout) and a theoretical infinite number of hands. For a counter with a bet spread, the calculation is more complex because the bet size varies. The practical approach is to use a simulation-based tool or our edge calculator, inputting effective edge and effective variance for your specific spread. The spreads and bankroll lesson applies these formulas to specific UK game conditions.

SCORE and DI: Standardised Comparison Metrics

Schlesinger's SCORE (Standardised Comparison Of Risk and Edge) provides a single number comparing the expected profit per 100 rounds for a counter with a fixed risk of ruin. It normalises edge, variance, and bet spread into one comparable metric, allowing different rule sets, games, and counting systems to be compared on an equal footing. A higher SCORE means more expected profit per 100 rounds at equivalent risk. The Desirability Index (DI) is a related per-unit-bankroll measure. Both are defined and tabulated in Blackjack Attack (3rd ed.) for a wide range of game conditions.

For a UK six-deck S17 game with 75% penetration and a 1-to-12 spread, the SCORE is approximately 30-35 in Schlesinger's framework, which is a reasonable result (higher-penetration games and fewer decks produce much higher scores; single-deck can reach 120+). The Hippodrome's main floor and Aspers Westfield Stratford both sit in this range: six decks, S17, 3:2, penetration around 70-75%. The bankroll calculator implements a simplified version of this comparison for common UK game configurations.

Key numbers

Sources: our in-house edge analysis, Blackjack Forum Online, Schlesinger Illustrious 18.