Introduction

Keno feels like a simple guessing game, but its mathematics are precise. Every hit, payout, and long-run expectation can be described with probability formulas. This guide breaks down the essentials: how hit probabilities are calculated, how RTP (Return to Player) connects paytables to outcomes, and why variance explains the “feel” of volatility in different variants.

Core Probability Model

Keno draws are modeled with the hypergeometric distribution. Out of 80 numbers, you select n spots. The game draws 20 numbers. The probability of hitting exactly k is:

p(k | n) = C(n, k) × C(80 – n, 20 – k) / C(80, 20)

Where C(a, b) is the binomial coefficient “a choose b.” This formula underlies all payout calculations and ensures results match theoretical predictions.

Expected Value (EV)

To connect probability to paytables, multiply the probability of each hit by its payout. Summing across all possible hits gives the expected return for a single bet.

EV = Σ [ p(k) × Pay(k) ]

If EV = 0.93, it means the game returns 93% of wagers in the long run (RTP = 93%). The remaining 7% is the house edge.

Return to Player (RTP)

RTP is the long-run percentage of wagers returned as winnings. It depends only on the paytable and the laws of probability, not on lucky streaks or player choices. Typical keno RTP ranges between 85% and 95% depending on jurisdiction and variant.

Variants (Cleopatra, Power, Lightning, etc.) rebalance paytables so their RTP falls within the same target range, even though volatility changes drastically.

Variance and Volatility

Variance measures how much outcomes swing around the mean RTP. Games with frequent small pays (e.g., Classic 4-spot) have low variance. Games with multipliers or large jackpots (e.g., Lightning, Power) have high variance. Variance per bet is:

σ² = Σ [ p(k) × (Pay(k)²) ] – RTP²

Larger variance = bigger emotional swings and deeper drawdowns, even if RTP is identical.

Simulation vs Theory

Long-run simulations confirm the math. After 100,000+ draws, observed hit frequencies match hypergeometric probabilities. RTP converges to the theoretical value. Variance bands appear clearly.

Practical Uses

• Estimate bankroll requirements for a given spot count.
• Compare paytables across casinos or variants.
• See why myths about “due numbers” or “hot streaks” fail.
• Understand why entertainment value comes from variance, not profit potential.

Worked Example: 6-Spot Table

Suppose you play a 6-spot table with the following payouts:

• 3 hits → 2×
• 4 hits → 5×
• 5 hits → 50×
• 6 hits → 1,500×

Using the probability formula, calculate p(3), p(4), p(5), p(6). Multiply each by the payout and sum. The result is RTP. Simulation will confirm it within ±0.1% over 100,000 rounds.

Summary

• Hit probabilities follow the hypergeometric distribution.
• EV = Σ probability × payout; RTP is EV as a percentage of stake.
• Variance explains volatility feel, not RTP.
• Simulations confirm math and expose bankroll drawdowns.

Next Steps

• Variant Breakdown: Cleopatra, Caveman, Power, Lightning — see how mechanics shift variance while RTP stays stable.
• Simulation Tools — run large draws to test theory against outcomes.
• Strategies & Myths — apply math insights to practical play.