How mathematical simulations work in iGaming

Mathematical simulations in iGaming are "virtual spins/bets" with the same rules as in the real game. You set the distribution of outcomes, describe the mechanics of bonuses and the player's strategy, and then thousands and millions of runs show how session results are distributed: average (EV), quantiles, frequency of "pluses," depth and duration of drawdowns. The simulation does not predict the next spin - it measures the distribution of what can happen at a distance.

1) What the simulation consists of

1. Step game model. Random variable (X) - multiplier to bet: 0; 0. 2; 1; 5; 10; … and their probabilities (p_j) (sum = 1).

2. Bonus mechanics. Freespins, sticky wilds, hold & spin, wheels/trails - often require states and transitions.

3. Player strategy. Bet size and stop rules: flat, progressions, profit/stop loss, pauses after L-series.

4. Session. Either fixed (N) spins or exit conditions (bank ≤ stop loss; achieved a break-profit; time limit).

The main thing: the strategy changes the form of the distribution of results, but not the very probabilities of fair play outcomes.

2) Two levels of allocations: "step" and "session"

Pitch (spin/bet). Gives the EV of one bet (\mu =\mathbb {E} [X]) and its variance (\sigma ^ 2).

Session. Sum of independent (or nearly independent) steps modified by bet/exit rules. Here they are interested in:

EV sessions;
quantiles of the result (Q50, Q75, Q90, Q95);
Target chance (e.g. ≥0% or ≥+20%)
max drawdown and its duration;
intervals of waiting for "significant" events (≥×10, bonus).

3) How to simulate: from simple to complex

A) Monte Carlo by "distribution passports"

Specify the pay baskets (≤×1, × 1- × 5, × 5- × 20, ≥×20) and their probabilities.

Generate random (U\sim [0,1]) and map to (X) via cumulative.

Apply the strategy: update the bank, count the metrics.

B) Markov processes

Sticky mechanics and multiplier upgrades make the outcome of the current spin state-dependent.

Status: (configuration of wilds/multiplier/spins remains).

Transitions: probabilities to the next state.

Reward: Expected winnings in step.

Consider the expectation and odds of thresholds a bottom-up iteration across states.

C) Hybrids

Model part of the round (bonus) as a Markov block, the basic game as independent steps. Combine.

4) Why "a lot of runs" is important

Slots have "heavy tails": rare very large payments give a significant part of EV. In small samples, they simply do not occur, and the estimate shifts.

For body picture: 10-50 thousand sessions of 1,000 spins.

For tail stability: 100k + and/or stratification (separate "if ≥×200 happened" scenarios).

See stability: double the number of runs - the metrics should hardly change.

5) What exactly to measure

EV sessions and the median result (the player "lives" the median, not the expectation).

Quantiles of result and drawdown (Q50/Q90).

Target chance (≥0%, ≥+20%).

Risk of ruin (probability of "zero "/stop loss before the completion of the plan).

Waiting intervals to ≥×10 and bonus (median, 75th percentile).

Sensitivity to session length and rate (heat maps).

6) Correct comparison of strategies

Common random numbers (CRNs). Run strategies on the same set of random outcomes. So you compare exactly the logic of bets, and not "luck."

Permutation tests and session pair bootstrap give the difference interval and (p) -value with no assumptions of normality.

Uniform success criteria in advance: for example, "90th percentile of drawdown ≤ 300 bets at a chance of ≥0% at least 40%."

7) Variance reduction

CRN - basic must-have.

Antithetic samples: pairs (U) and (1-U) reduce noise.

Stratification: Separately simulate rare large events and weigh.

Aggregation by baskets: instead of a detailed table of payments - 4-6 intervals, almost the same risk picture, but faster.

8) Reproducibility and honesty of the experiment

Fix the RNG seed and keep the model versions.

Don't change the rules as you go. Any adaptation "after seeing the data" makes the result invalid.

Same noise for A/B. Otherwise, the difference is often phantom.

Reports at intervals. The average without confidence bands is an invitation to self-deception.

9) Where simulations are particularly useful

Complex bonuses: stairs, multiplier progressions, sticky mechanics.

Bonus purchase: (EV_{\text{net}}=\mathbb{E}[X]-C) and comparison of risk profile "buy" vs "natural input."

Rate management: how much does the progression "cost" in terms of Q90 drawdown and a chance of ≥0%.

Session plan: how the chance of goals changes with spin 200/500/1000.

10) Typical errors

Small volume with heavy tails → "strategy dragged" by accident.

Mixing different RTP versions/slots in one experiment.

The test "to the first plus" is a strong bias.

Absence of CRN - comparison on different "noise."

Conclusions on the average without quantiles/drawdowns - ignoring the real risk.

11) Simulation mini pseudocode


input: {x_j, p_j} - pitch distribution; B0 - start-up bank; N - spin plan; S - repeat strategy M times:
B:= B0; peak:= B; maxDD: = 0 for t = 1.. N:
x: = case of {x_j, p_j}
bet: = bet _ rule (B, t, history, S)
win:= bet x
B:= B + (win - bet)
peak:= max(peak, B); maxDD:= max(maxDD, peak - B)
if conditions S require stop/pause → exit the cycle save total (B-B0), maxDD, duration, event counters after M runs: count EV, quantiles, risk, waiting intervals for strategy comparison - same x (CRN), bootstrap/permutation for difference

12) Limitations and ethics

Simulations do not turn negative expectation into positive expectation; they show the price of volatility.

Real stocks/cashback/tournaments change the final economy - consider them separately.

The psychology of real money differs from the demo: transfer the rules of limits and pauses to the combat game.

When publishing the results, show the technique, RNG seed and volumes - this is protection against cherry-picking.

Bottom line: simulations are an iGaming laboratory: you formalize mechanics, honestly "spin" virtual sessions and get not myths about "timing," but verifiable numbers - EV, quantiles, drawdowns and the risk of ruin. With the right formulation (CRN, large volumes, uncertainty intervals), simulations provide a reliable basis for decisions about rates, limits, session duration and choice of volatility.