Core Model

All keno variants are grounded in the same combinatorial model. There are 80 numbers. You pick n. The game draws 20 without replacement. The probability of seeing exactly k hits is determined by the hypergeometric distribution:

P(K=k | n) = [C(n,k) * C(80−n, 20−k)] / C(80,20)

This distribution is the backbone of every calculation. Variants (Cleopatra, Caveman, Fireball, Lightning, Power, DaVinci, etc.) change how payouts are mapped to outcomes, but the underlying probabilities of K hits do not change.

Expected Value (EV) and RTP

Expected value and return-to-player (RTP) quantify the long-run mean of the payout distribution. For a paytable Pay(k) that returns credits per $1 bet when you hit k numbers:

RTP = Σ_k P(K=k) · Pay(k)

EV = RTP − 1

House edge = 1 − RTP

RTP is dimensionless (credits returned per credit wagered). EV is the average net change per $1 staked. House edge is the fraction withheld by the paytable.

Variance and Volatility

Variance measures the spread of possible results around the mean. It grows as jackpots or multipliers become steeper.

σ² = Σ_k P(K=k) · Pay(k)² − RTP²

Larger σ² means more dramatic bankroll swings. High-variance games offer big jackpots but long droughts; low-variance games produce steadier, smaller payouts.

Bankroll Planning (Rule-of-Thumb)

A practical question: How large should your bankroll be to withstand swings? Two simple guides:

1. Drawdown-based: Use simulator logs. Start bankroll ≈ 8–12× your average drawdown for target session length.
2. Kelly fraction (approximate):

   f* ≈ edge / σ²

   Use this only with caution; edge is negative in most keno, so this guides risk sizing rather than positive growth.

The principle: variance scales with risk-of-ruin. If you want ≤5% chance of ruin in T rounds, budget accordingly.

Interpreting Simulations

Keno draws are independent and identically distributed under the RNG. That means law of large numbers applies: long runs converge to theoretical RTP. To quantify precision:

95% CI after N games = RTP ± 1.96·σ / √N

Use this to decide how many rounds you need. For variance-heavy variants, more rounds are required for stable estimates.

Practical Workflow

1. Pick variant and paytable.
2. Run 100k–1M draws.
3. Record RTP, σ, drawdown, and hit distribution.
4. Adjust spots/paytable. Repeat.

Iteration builds intuition. Seeing charts of payout histograms and drawdown distributions conveys more than formulas alone.

Notes by Variant

• Classic: Lowest variance at given RTP. Pure hypergeometric.
• Cleopatra: Free games raise variance; clustered outcomes.
• Power: Last-ball multiplier inflates right tail; variance higher.
• Lightning: Random multipliers; extreme variance.
• Caveman: Egg multipliers; variance depends on multiplier table.
• Fireball: Substitution raises hit frequency; variance depends on upgrade caps.
• DaVinci: Wild substitutions smooth near-misses; variance profile between Fireball and Caveman.

FAQ