Educational worksheet. Why a 1.06% theoretical Baccarat Banker house edge cannot predict a 100-hand session, with an approximate variance calculation.
Two different questions: average and spread
House edge answers what a wager loses on average over a very large number of repetitions. Volatility describes how widely individual outcomes can fluctuate around that average. A 1.06% house edge on a standard eight-deck Banker wager is not a promise that someone who risks $1,000 will finish exactly $10.60 down. Individual hands mostly create much larger positive or negative movements. To separate the two, construct a simple model with three net results for a one-unit Banker wager: +0.95 on Banker, −1 on Player, and 0 on Tie.
Expected result on one hand
Using a fresh eight-deck reference distribution, P(Banker) = 0.458597, P(Player) = 0.446247 and P(Tie) = 0.095156. Expected result E[X] = 0.458597 × 0.95 + 0.446247 × (−1) + 0.095156 × 0 ≈ −0.01058 units. That is a negative long-run average of around 1.06% per unit staked. If the stake were $10 and the number of rounds were 100, the model’s expected net change would be 100 × $10 × (−0.01058) ≈ −$10.58. This arithmetic assumes an unchanged stake for every round; it is not a betting recommendation.
Why actual results can depart so far
A useful statistical measure is variance. First calculate E[X²] = 0.458597 × 0.95² + 0.446247 × (−1)² + 0.095156 × 0² ≈ 0.8601. Subtract E[X]² (around 0.000112) to obtain a one-hand variance of about 0.8600 and a standard deviation close to 0.927 units. The units matter: this is variation measured in full wager sizes, vastly larger than the tiny expected loss per hand. A single Banker loss is −1 unit regardless of the 1.06% headline.
An approximate 100-hand spread
If we approximate successive deals as independent repetitions of the same distribution, total variance over 100 hands is 100 × 0.8600, and standard deviation is its square root: about 9.27 units. With fixed $10 example stakes, that is approximately $92.70 of standard deviation around an expected loss of $10.58. Standard deviation does not mean a guaranteed gain or loss range, and an actual finite shoe creates some dependence between hands. This example’s purpose is to demonstrate scale: expected value and short-session outcome are entirely different questions.
Why more hands do not make a loss predictable
Under a stationary independent approximation, expected total loss grows proportionally with the number of rounds, while the standard deviation grows approximately with its square root. The average per round becomes more stable in a large sample, but the potential for a large absolute financial loss remains. Faster games can accumulate turnover more quickly; they do not improve the payoff mathematics. Neither a favourable short run nor an unfavourable one establishes that the theoretical probabilities changed.
Limits of the demonstration
Actual sessions may involve changing stakes, pushes, different commission settlement, continuous-shuffle procedures or variant-specific bets. Those changes require different calculations. The model does not estimate a particular individual’s chance of finishing ahead, nor should it be used to justify increasing exposure. Treat the expected-value formula as a way to interpret risk language, not as a plan for spending.
Related lessons
RTP, house edge and turnover · Standard main-bet probabilities
Sources and methodology
Wizard of Odds — standard Baccarat probabilities; Wizard of Odds — gambling session calculator. Worked examples are hypothetical, not recorded casino results or predictive systems. Figures are educational estimates for stated rules, not live odds.
For adults in jurisdictions where gambling is legal. Learning the rules does not reduce the financial risks of gambling; stop if it causes harm.