BROWNIAN MOTION // STOCHASTIC DISPERSION

Session Variance Simulator — Bell Curve & Risk of Ruin

Q: What is the N-Zero (N₀) metric in blackjack?

N-Zero is the number of hands required for cumulative expected value to equal exactly one cumulative standard deviation (N₀ = σ² / EV²). Once a player logs N₀ hands (typically 25,000–35,000 in shoe games), the probability of having a net positive profit is 84.1%.

Q: Why does variance expand dramatically under aggressive bet spreads?

Because wagering 12 to 16 units on favorable hands concentrates statistical volatility. Losing a maximum bet wipes out 15 minimum bets, increasing variance from flat 1.32 to ~3.2 units squared per round.

Q: How does bankroll size relate exponentially to Risk of Ruin?

Under Brownian motion models, RoR = exp(-2*B*EV/σ²). Because bankroll sits in the negative exponent, doubling the bankroll squares the safety factor (reducing a 10% ruin risk to 1.0%).

Model session volatility, confidence intervals (±1σ, ±2σ, ±3σ), and mathematical Risk of Ruin across varying bankroll depths and bet spreads.

SIMULATION PARAMETERS EXPONENTIAL ROBUSTNESS

Bankroll Size (Units of Min Bet)

Unit Monetary Value ($)

Bet Spread Architecture

Session Sample Size (Hands)

Blended Expected Advantage (%)

+1.0%

Gaussian Outcome Dispersion (Bell Curve)

NORMAL APPROXIMATION

68.2% Range (±1σ) -$1,250 to +$1,750

95.4% Range (±2σ) -$2,750 to +$3,250

99.7% Range (±3σ) -$4,250 to +$4,750

RISK METRICS CONTINUOUS MODEL

LIFETIME RISK OF RUIN (RoR) 0.8% Institutional Safety (< 2%)

SESSION WIN PROBABILITY 56.4%

EXPECTED MEDIAN PROFIT +$250 +10.0 units

N-ZERO (N₀) HANDS HORIZON 32,000 hands ~320 hours table play

Frequently Answered Questions

What is the N-Zero (N₀) metric in blackjack? ▼

N-Zero is the number of hands required for cumulative expected value to equal exactly one cumulative standard deviation (N₀ = σ² / EV²). Once a player logs N₀ hands (typically 25,000–35,000 in shoe games), the probability of having a net positive profit is 84.1%.

Why does variance expand dramatically under aggressive bet spreads? ▼

Because wagering 12 to 16 units on favorable hands concentrates statistical volatility. Losing a maximum bet wipes out 15 minimum bets, increasing variance from flat 1.32 to ~3.2 units squared per round.

How does bankroll size relate exponentially to Risk of Ruin? ▼

Under Brownian motion models, RoR = exp(-2*B*EV/σ²). Because bankroll sits in the negative exponent, doubling the bankroll squares the safety factor (reducing a 10% ruin risk to 1.0%).