1) Introduction

Cleopatra Keno augments Classic Keno with a single, high-leverage twist: if the 20th and final ball drawn is one of your selected numbers, the game awards a set of free games (most commonly 12) in which all wins pay 2×. That final-ball trigger concentrates volatility into a single event that can swing a session’s trajectory. The base game remains Classic Keno—pick numbers, draw 20 without replacement—yet the free-game package overlays an additional expectation and a distinct rhythm to bankroll flow.

This chapter formalizes rules, provides representative paytables, derives trigger probabilities, and decomposes expected value (EV) into base and bonus components. We examine how spot count (n chosen numbers) interacts with the final-ball trigger and why mid-spot counts often balance time-on-device with bonus exposure. We then translate the math into strategy templates, supply graph placeholders, and outline simulation methodology so you can reproduce results with your own paytables and seeds.

Use Cleopatra Keno to study how a conditional multiplier mode changes session profiles without altering base-draw combinatorics. This knowledge generalizes to other “mode-switch” variants that hinge on sporadic triggers driving streaks of elevated EV.

2) Rules & Gameplay

2.1 Core Rules

• Board: Numbers 1–80.
• Draws per round: 20 unique numbers sampled without replacement.
• Your selection (“spots”): Typically 1–10 for analysis; some venues offer up to 15 or 20.
• Stake: Base wager per round (e.g., 1 credit). Multi-card and multi-credit options are common.
• Base payouts: Determined by a published paytable mapping hits to prizes.
• Cleopatra trigger: If the final ball (the 20th) is one of your selected numbers, you are awarded 12 free games (common), where all wins pay 2×.
• Free games: Play like base rounds, usually at the same spot selection, without deducting stake per round. No additional stakes are consumed during the free-game series.
• Retriggers: Many implementations do not retrigger within free games. If your venue supports retriggering, the math and variance increase; update formulas accordingly.

2.2 Round Walk-Through

1. Select a spot count, e.g., 6, and mark six numbers.
2. Place a stake (1 credit).
3. Draw 20 numbers. Tally hits and consult the 6-spot paytable for base payout.
4. Check the final ball. If it is among your six numbers, award 12 free games at 2× pays.
5. Resolve the 12 free games: each consists of a new 20-ball draw with the same marked numbers; each win pays double; no stake is deducted; retrigger typically disabled.
6. Net result = base payout + sum of free-game payouts − base stake.

2.3 Design Implications

The final-ball trigger creates a mode switch from base EV to a higher-EV burst across a fixed number of rounds. Bankrolls experience clusters of returns rather than evenly distributed mid-tier pays. The player’s experience is defined by two coupled distributions: the hypergeometric hit distribution per round and the Bernoulli-like success of the final ball matching any of your picks.

3) Paytables

Paytables vary by operator and jurisdiction. The examples below are pedagogical. Replace with your actual venue’s tables for precise calculations. In Cleopatra free games, payouts are typically doubled relative to base.

3.1 Example 4-Spot Base Paytable

Free games: same mapping but 2× the payout per hit.

3.2 Example 6-Spot Base Paytable

Free games: 2× these payouts. EV uplift applies across the 12 free rounds when triggered.

3.3 Example 8-Spot Base Paytable

Free-game mode doubles all wins. Variance increases due to clustered 12-round bursts.

3.4 Practical Use

• Compute base EV from the paytable and hypergeometric hit probabilities.
• Compute free-game EV per round as 2× base EV (same spots) unless the venue uses altered free-game tables.
• Total EV = base EV − stake + trigger probability × (12 × free-round EV).

4) Mathematics of Cleopatra Keno

4.1 Hit Probability per Round

Let n be your spot count and K the random number of hits in a 20-ball draw from 80. As in Classic Keno, hits follow the hypergeometric distribution:

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

4.2 Final-Ball Trigger Probability

The final ball is uniformly one of the 80 numbers, independent of which numbers appeared earlier in the round except that it is distinct from them (no replacement). For the purpose of “is the final ball one of my n picks,” symmetry yields:

P(trigger | n spots) = n / 80

This result is intuitive: before any draws, each number is equally likely to occupy the 20th position. With n picks, n of the 80 possible final numbers are favorable. This probability holds for the base round. If retriggers are allowed inside free games, apply the same logic for each free round.

4.3 Expected Value Decomposition

Define:

• EV_base_round = Σ_k P(K=k) × pay_base(k)
• EV_free_round = Σ_k P(K=k) × pay_free(k)

If free games pay exactly 2× the base paytable, then EV_free_round = 2 × EV_base_round.

With T free games per trigger (commonly 12) and p = n/80:

EV_total_per_base_round
  = (EV_base_round - stake)
    + p × (T × EV_free_round)
    = (EV_base_round - stake) + (n/80) × (T × EV_free_round)
    [if EV_free_round = 2 × EV_base_round]
  = (EV_base_round - stake) + (n/80) × (T × 2 × EV_base_round)
  = (EV_base_round - stake) + (nT/40) × EV_base_round
  = EV_base_round × (1 + nT/40) - stake
    

This decomposition isolates how spot count n scales bonus contribution linearly through trigger probability, while T scales it via free-game length. If your venue’s free-game paytable differs from 2×, substitute the actual EV_free_round.

4.4 Variance and Burstiness

Variance grows from two sources: base hit dispersion and bonus clustering. The latter introduces autocorrelation in outcomes because 12 free rounds arrive consecutively. For short sessions, these clusters dominate perceived luck. Analytically, total variance per base round can be expressed via the law of total variance over the Bernoulli trigger, but practically, simulation is the superior tool to visualize bankroll paths.

4.5 Worked Example (Illustrative)

Suppose a 6-spot base EV of 0.47 credits per 1-credit stake under a given table. With T = 12 and n = 6:

EV_total ≈ 0.47 × (1 + nT/40) - 1
        = 0.47 × (1 + 72/40) - 1
        = 0.47 × (2.8) - 1
        = 1.316 - 1
        = 0.316 credits per base round
    

That is a stylized example to demonstrate mechanics, not a claim about any specific venue’s RTP. Real EV depends on actual paytables.

5) Graphs & Charts

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6) Strategy Insights

6.1 Define Objectives

• Time on device: Favor 3–5 spots. Base pays appear with some regularity and triggers occur at manageable intervals (p ≈ 3/80 to 5/80 per base round).
• Balanced peaks: 6–7 spots. Trigger rate rises, free-mode value climbs, and mid-tier hits help sustain sessions.
• Jackpot leaning: 8–10 spots. Higher trigger probability and large top-end pays, but more dead base rounds. Bankroll variance rises.

6.2 Spot Count vs Trigger Economics

Because p = n/80, each additional spot increases trigger chance by 1/80 per base round. The marginal value of +1 spot equals (T × EV_free_round)/80. If your free-round EV is high, the incentive to add spots grows, tempered by the base table’s variance. Evaluate both base play feel and bonus economics.

6.3 Bankroll Management

• Unit sizing: Target 200–300 base rounds at your chosen volatility. Bonus clusters will induce spikes; do not raise unit size mid-cluster.
• Stop-loss & locks: Pre-commit to a max drawdown. When a bonus series lands well, lock a portion and drop back to baseline unit.
• Expectation realism: The average time to a trigger is 80/n base rounds. Plan sessions accordingly.

6.4 Paytable Sensitivity

If free-game payouts differ from 2×, recompute EV_free_round. Small increases in free-round EV have amplified effect because they multiply by T and the trigger rate. Seemingly minor free-table tweaks can materially shift RTP and volatility.

6.5 Myths to Ignore

• “Saving” the trigger: The final ball is random. There is no memory to the process.
• Hot/cold numbers: Under a fair RNG, frequency fluctuations are noise, not signal.
• Board patterns: Cosmetic. EV is governed by tables and trigger math.

6.6 Practical Templates

• Low-variance Cleopatra: 4-spot, 1 credit, 300 base rounds. Expect moderate trigger cadence and manageable swings.
• Balanced Cleopatra: 6-spot, 1 credit, 250 base rounds. Periodic free-series deliver memorable peaks.
• Aggressive Cleopatra: 8–10 spots, 1 credit, 200 base rounds. Plan for droughts; bank on clustered upside.

7) Simulation Results

Simulations validate analytic expectations and reveal path characteristics. Use at least 500k–1M base rounds per configuration to stabilize RTP and capture tail events. Report seeds, spot count, paytables, free-game rules, and whether retriggers are allowed.

7.1 Methodology

• Independent base rounds with 20 draws from 80 without replacement.
• Trigger test on final ball per round; if success, play T free rounds at 2× pays, no retrigger.
• Metrics: RTP estimate, standard error, variance per round, trigger rate, average free-series value, time-to-trigger distribution, drawdown quantiles, and cumulative return percentiles.

7.2 Expected Qualitative Findings

• RTP convergence: Decomposition predicts simulation means accurately when paytables are correct.
• Burstiness: Bankroll series show flat segments between triggers punctuated by step-ups during 12-round bursts.
• Spot sensitivity: Trigger rate scales with n. High-spot games exhibit larger jumps but deeper intervening troughs.
• Drawdown tails: Longer right-tail rewards are offset by longer droughts; risk limits should reflect that asymmetry.

7.3 Example Output Tables (Placeholders)

8) Case Studies

8.1 Low-Variance Cleopatra (4-Spot, 300 Base Rounds)

A 4-spot player stakes 1 credit per base round. Triggers arrive on average every 20 rounds. Most base rounds are non-paying or small pays, but free-series inject double-payout opportunities. The session feels steadier than higher-spot play, yet clustered wins still define the day’s peaks.

8.2 Balanced Risk (6-Spot, 250 Base Rounds)

At 6 spots, triggers average every ~13 rounds. The player experiences stretches of quiet punctuated by energetic 12-round bursts. Mid-tier base hits keep morale afloat between bursts; when a burst lands, the 2× multiplier converts mid-tiers into strong segments.

8.3 Aggressive Profile (8–10 Spots, 200 Base Rounds)

The player hunts large, doubled wins. Triggers are more common than low-spot play, but dead base rounds increase. If bursts underperform—due to variance in free rounds—the outcome can still be negative. This profile suits players comfortable with deep drawdowns seeking large positive skew.

8.4 Operational Constraints

Some venues apply slightly different free-round tables or cap certain wins during free mode. Always verify the implementation. Small rule differences propagate through EV and risk.

9) FAQ: Practical Questions

9.1 Does the order of the first 19 balls matter for the trigger?

No. Only the identity of the 20th ball matters for awarding free games.

9.2 Can free games retrigger?

Often no. If yes, use the same p = n/80 per free round and recompute EV with a geometric sum for expected extra series.

9.3 Do I need more spots to “aim” for the trigger?

More spots increase p linearly, but also change base volatility. Choose n balancing bonus exposure and base feel.

9.4 Are numbers’ past appearances predictive?

No. Under fair RNG, outcomes are independent from round to round.

10) Summary & Takeaways

• Cleopatra Keno = Classic Keno + final-ball trigger into 12 free games at 2×.
• Trigger probability p = n/80. Spot count linearly scales bonus opportunity.
• Total EV = base EV minus stake plus expected value from free-series.
• Variance rises due to clustered outcomes. Bankrolls must absorb droughts.
• Pick spot counts by objective: time on device (3–5), balanced peaks (6–7), or aggressive skew (8–10).