The mean is not the outcome

Public casino-bonus calculators answer one question (what is this bonus worth?) with one number, computed one way: expected value = bonus − (required turnover × house edge). That number is a mean. This study asks what the rest of the distribution looks like, and whether the mean is a fair summary of it.

Six bonus structures, read on 6 September 2026 from operators licensed in US states that permit online casino play. Five distinct sets of terms, not six: two of the structures are the same 100% match, 15x playthrough offer at two different completion windows, 14 days and 7 days, and both are in the published grid and the full result set. The tables below report the 14-day one, since the two behave identically. We kept both rather than dropping one, because two independent 60,000-attempt runs of the same terms landing on the same loss count at every stake is itself a useful check that the engine is stable, and because the 7-day window never binds ahead of the 14-day one at any stake in the sweep, so the pair is a robustness check rather than a second finding.

This sample is small, and one entry in six is verified against the operator's own published terms. The other five are as reported by third-party coverage, because the operators' own pages returned HTTP 403 to automated access or gated by state. Every source below carries its verification status, and every statement in this piece that rests on secondary terms says so. This is the single biggest limitation of the study and it is not in a footnote.

One observation from the single offer where both primary and secondary text were obtained: the operator's own terms carried a 60-plus-game exclusion list and four separate expiry clocks, 3, 7, 14 and 30 days, each governing a different part of the offer. None of that appeared in any secondary summary of the same offer. We have one case, not a pattern, and state it as one; but it suggests the terms that cost a player the most are the ones least likely to survive into second-hand coverage.

Deposit and bonus are held at $100/$100 throughout. Every offer sampled is a 100% match whose published cap is well above $100, so this is a real, claimable instance of each. Holding them constant means any difference in the results is attributable to the terms and to nothing else.

We could not verify a maximum-bet clause for any offer in the sample. Maximum bet while a bonus is active is a well-known term, and the model supports a cashout cap, but nothing in the accessible sources gave us a figure we would stand behind. It is therefore absent from the model, and any offer that carries one will perform worse than modeled here.

A spin returns nothing with probability 0.75, and otherwise pays a multiple of the stake drawn from a two-part mixture: a body of ordinary wins (bounded Pareto on 0.2× to 20×) and a tail of rare large wins (bounded Pareto on 20× to 10,000×, shape 1.6).

Why a mixture: a single power law that returns 96% off a 25% hit rate has to be so heavy-tailed that it pays 100×-or-better roughly once every 980 spins, which no real game does. Two components fix that. At the calibrated settings the modeled games pay 100×-or-better once per 374,000 spins at the steady end and once per 1,190 spins at the swingiest, a realistic spread.

The mixture weight and body shape are solved (by bisection, to twelve significant figures) so that each profile hits its stated RTP and its stated standard deviation exactly. They are not hand-tuned.

A caveat we want to be explicit about. No operator or game supplier publishes a standard volatility figure, and there is no industry-standard definition of “high volatility.” The labels attached to the five profiles are orientation only and should not be read as claims about where any real game sits. What this study reports is the shape of the relationship across the sweep, which does not depend on where any particular game falls on it.

The engine carries 35 tests, run in continuous integration alongside the rest of the site.

The load-bearing one is a closed-form check. Every dollar staked loses the house edge in expectation, and the bonus is a straight credit, so the mean net outcome must equal bonus − edge × (mean amount staked), exactly, regardless of how the wagering is structured or how the player busts. The simulation is asserted against that identity at three different multipliers, to within its own reported standard error. If the engine were leaking or creating money anywhere, that test would fail.

The rest check that the generator is deterministic and in range; that the bounded-Pareto mean and shape solvers invert each other; that each game profile hits its stated RTP analytically and its stated volatility exactly; that sampled returns stay inside the stated win bounds; that a zero-wagering bonus is kept in full; that an all-losing game busts in exactly the arithmetic number of spins; that game weighting multiplies required turnover; that a cashout cap binds; that clearing gets monotonically less likely as the multiplier rises; and that percentiles come back ordered.

A further test holds the figures on this page against the study's own output file, so a published number cannot drift from the data behind it.

60,000 simulated attempts per scenario; 180 scenarios (6 structures × 5 game profiles × 6 stakes).

The mean net outcome is the least reliable statistic in this study, and the results report its standard error next to it for that reason. It is carried by rare very large wins, so it converges slowly, the clear rate and the median are far more stable at the same trial count.

That is not a limitation to apologize for. It is the finding: the one number every public calculator reports is the one that needs the most samples to pin down and describes the fewest players.

The run is seeded and deterministic, it uses a mulberry32 generator rather than the platform's random source, so the same engine and inputs produce the same results file byte for byte on any machine. That determinism is what lets every figure on this page be checked automatically against the result set the run produced, not a claim that the run can be repeated from this page alone.

The engine, the sampled terms and the 35-test validation suite are kept as the study's working files rather than published here. What is published below is a CSV summarizing the full 180-scenario grid at the precision this page's tables use, one row per scenario. It is not the engine's raw per-trial output: a few columns from the underlying run, the wider percentile spread, the probability of finishing ahead outright, and per-attempt staking detail, stay in the working files rather than the CSV. If you want to check a specific number against the underlying run, or want the engine and validation suite made available to verify the whole thing yourself, see the corrections policy: there is no dedicated intake for that yet, and this note says so rather than pointing at one that does not exist.