A $100 Banker wager does not lose about $1.06 each hand. It usually wins $95, loses $100, or pushes. The small house edge appears only after those large outcomes are averaged over many decisions.
That gap between the average and the individual result is variance. It explains why baccarat can produce severe short-term swings even when the main wagers have comparatively low house edges.
Start with the outcome distribution
For a standard eight-deck commission game, commonly cited probabilities are approximately:
- Banker wins: 45.8597%;
- Player wins: 44.6247%;
- Tie: 9.5156%.
A Tie normally pushes ordinary Banker and Player wagers. With 5% commission, one unit wagered on Banker has these net outcomes:
| Result | Probability | Net result per unit |
|---|---|---|
| Banker wins | 0.458597 | +0.95 |
| Player wins | 0.446247 | -1.00 |
| Tie | 0.095156 | 0.00 |
The expected value is:
μ = Σ pᵢxᵢ
where pᵢ is the probability of outcome i, xᵢ is its net result, and Σ means add all outcome contributions.
For Banker:
μ = (0.458597 × 0.95) + (0.446247 × -1) + (0.095156 × 0)
μ ≈ -0.01058 units per wager
That corresponds to an expected loss of about 1.06% of the amount wagered.
Variance measures the bumpiness
Variance is calculated as:
Variance = Σ pᵢ(xᵢ − μ)²
The standard deviation is the square root of variance:
Standard deviation = √Variance
For the Banker outcome distribution above:
- expected value: about -0.01058 units;
- variance: about 0.86002 unit²;
- standard deviation: about 0.92737 units per hand.
The standard deviation is nearly one full betting unit, while the expected loss is only about one-hundredth of a unit. That is the central fact of baccarat session risk.
What 100 hands can look like
For an approximate fixed-bet model with independent hands:
Expected result after n hands = nμ
Standard deviation after n hands = σ√n
where n is the number of hands, μ is expected value per hand, and σ is standard deviation per hand.
For 100 Banker wagers of $100:
Expected result
100 × -0.01058 × $100 ≈ -$105.80
One standard deviation
0.92737 × √100 × $100 ≈ $927.37
The ordinary swing scale is therefore much larger than the expected loss. A result hundreds of dollars ahead or behind does not disprove the house edge. It is exactly the kind of variation the distribution permits.
This is an approximation. Cards are dealt without replacement, so hands within a shoe are not perfectly independent. Commission methods and rule variants also change the outcome values. The formula remains useful for understanding scale, not for predicting the next shoe.
Banker, Player, and Tie do not swing alike
Using the same eight-deck probabilities, approximate per-unit figures are:
| Wager | Expected value | Standard deviation | Main reason |
|---|---|---|---|
| Banker, 5% commission | -0.01058 | 0.92737 | Win pays 0.95; ties push |
| Player | -0.01235 | 0.95115 | Wins and losses are ±1; ties push |
| Tie, paying 8 to 1 | -0.14360 | 2.64087 | Rare +8 win, frequent -1 loss |
The Tie wager combines a much worse expectation with far greater volatility. A single hit can dominate a short session, which makes the wager feel successful even when repeated play is expensive.
The published outcome frequencies and house advantages are also reported in a peer-reviewed study of millions of land-based baccarat games; see the empirical baccarat gambling study. The calculations here convert those probabilities into variance and standard deviation for educational comparison.
Flat betting changes dollars, not unit variance
Flat betting one unit keeps the outcome distribution stable in unit terms. Changing the unit from $25 to $250 multiplies both expected loss and standard deviation by ten.
For 100 Banker hands:
| Unit size | Expected loss | Approx. one-SD swing |
|---|---|---|
| $25 | $26.45 | $231.84 |
| $100 | $105.80 | $927.37 |
| $250 | $264.50 | $2,318.43 |
A larger bankroll does not reduce variance. It gives the player more capacity to absorb the same unit swings. The baccarat bankroll-risk guide explains that distinction.
Progressions make the distribution unstable
A betting progression changes the amount at risk after previous results. The next hand still follows the baccarat rules, but the dollar outcome now depends on the sequence because the unit size is moving.
Suppose a player wagers $25, then doubles after each loss:
$25 → $50 → $100 → $200 → $400
Five consecutive losses cost $775. The progression does not change the probability of Banker, Player, or Tie. It concentrates more money on later hands and creates a heavily skewed session distribution.
Flat betting cannot turn a negative expectation positive, but it makes the relationship between hand variance and dollar variance easier to control.
Streaks are a consequence, not a signal
Runs of Banker or Player are visually striking because baccarat roads preserve sequence. A streak is compatible with random variation; it does not create a compensating force or a continuation guarantee.
After six Banker wins, the seventh hand is resolved by the remaining shoe composition and fixed drawing rules. The scoreboard does not enter the calculation. Treating a run as predictive often leads to larger wagers precisely when emotion is strongest.
The separate pages on the Banker streak myth and Player streak myth address those claims directly.
What variance should change in practice
Variance is not a reason to avoid mathematics. It is the reason to use it correctly.
- Compare wagers by both expected value and standard deviation.
- Set unit size from the loss you can absorb, not from the size of the last win.
- Include game speed when estimating total exposure.
- Do not interpret a short winning session as evidence of positive expectation.
- Do not interpret a short losing session as a debt the game must repay.
- Keep Tie and volatile side bets separate from main-wager analysis.
The variance simulator can show many possible paths with the same assumptions. Its purpose is not to forecast a result; it is to make the range of plausible results visible.
Baccarat's low main-bet edge describes the center of the distribution. Variance describes how far a real session can wander from that center before the long run becomes visible.