Crash Bust Distribution: Checking a Casino’s Curve Yourself

Crash games present one verifiable claim: the multiplier at which a round busts is a deterministic function of published seed material. If that claim survives scrutiny, the long-run distribution of bust points must follow a specific curve — usually 1/m or (1 – house edge)/m. This article walks through collecting a sample of rounds, replaying the seed-to-bust conversion, and comparing the observed survival curve to the theoretical one, all without relying on a casino’s declared RTP. A recurring theme in our news coverage is that statistical checks settle whether a game pays as advertised.

Where the bust point comes from

Most crash implementations reduce a provably fair hash to an integer h uniformly distributed over a 52-bit range. The operator documents how h is produced — typically an HMAC-SHA-256 over a server seed, client seed, and round number, or the terminal value of a previously committed hash chain. The conversion then follows one of two widely used families. The first, popularized by Bustabit and used in many clones, is:

h    = int(hash[:13], 16)          # first 13 hex chars = 52 bits
raw  = 2**52 / (2**52 - h)         # multiplier in [1, 2^52]
bust = max(1, floor(100 * raw) / 100)

The second embeds the house edge directly in the curve:

bust = max(1, floor(100 * (1 - HE) / (1 - h / 2**52)) / 100)

If a casino publishes its exact formula, anyone can implement it in a few lines of Python and replay any seed pair. What you cannot verify from a single round is whether the curve behaves as advertised; that requires a sample.

The curve you should see

Let h be uniform over [0, 2^52). In both families above, the unrounded multiplier M satisfies a simple survival law for thresholds m in [1, 2^52):

P(bust >= m) ≈ (1 – HE) / m

where HE is 0 for the Bustabit-style formula (the operator then applies a fee on winnings) and 1 or 2 percent for casinos that build the edge into the curve. The approximation error comes from flooring to two decimals and from the boundary at m = 1. The table below gives the theoretical survival fractions for the two most common parameterizations.

For the 1% edge case, about 50.5% of rounds bust below 2.00 and 80.2% below 5.00. The frequency of an exactly 1.00 bust also differs by parameterization: about 1.0% of rounds in the zero-edge curve, and about 2.0% when a 1% edge is embedded, because the survival tail above 1.01 is 0.99 / 1.01 ≈ 0.98. These boundary effects are why the reference curve should always be computed from the operator’s exact formula rather than from the 1/m shorthand.

Collecting a usable sample

The raw material is available wherever a casino publishes its provably fair history: on the game page, in a public API, or on-chain. Record the round identifier, the server seed / client seed pair, the published hash, and the bust multiplier. Before running any statistics, replay each round: regenerate the hash from the seeds and confirm your conversion returns the exact published bust. Also verify the seed chain — the round N+1 seed should hash to the value the operator committed before round N — otherwise the sample is not provably linked to the advertised randomness. For the mechanics of these commitments, see our provably fair guides.

Sample size matters more than the number of checks. At N = 10,000 rounds, the binomial standard error of an observed survival fraction at m = 2 is sqrt(0.5 * 0.5 / 10000) = 0.5 percentage points, so a 49.5% theory with an observed 48.5-50.5% band is unremarkable. For tail thresholds such as m = 10, only about 990 rounds fall in the survival set, and the relative noise is several times larger. Treat 10,000 rounds as a practical minimum for a mid-curve audit, and several tens of thousands if you care about m >= 10.

Comparing the observed busts to the curve

For each threshold m in {1.1, 1.5, 2, 3, 5, 10}, sort the sampled busts and count the fraction that bust at or above m. Plot the empirical survival function and overlay the theoretical curve. Two diagnostics catch most real deviations. First, the survival at the mid multipliers: a deficit of 200 basis points or more at m = 2.00 over N = 10,000 corresponds to about 4 standard errors, which is difficult to attribute to luck. Second, the shape of the tail: a long-run edge that appears only above m = 50 shows up as an overfull bucket of low busts and a thinning of the 10x-100x range.

For a rigorous pass/fail, group busts into eight buckets — [1.00, 1.10), [1.10, 1.20), [1.20, 1.50), [1.50, 2.00), [2.00, 3.00), [3.00, 5.00), [5.00, 10.00), and [10.00, inf) — and run a chi-square goodness-of-fit test on the observed counts. With N = 10,000, every cell has an expected count well above 5, and the test statistic should sit below the 5% critical value, which is 14.1 for 7 degrees of freedom. Also compute the lag-1 correlation of bust multipliers; a casino that shuffles a fixed profit pool typically shows non-zero autocorrelation in the lower end of the range. Any result that survives those two checks is consistent with a fixed, advertisable house edge. That measured edge, not the marketing RTP, is the number that belongs in a bankroll management model.

When the curve disagrees

Before accusing the operator of running a fake distribution, check the mundane explanations. The exact rounding rule matters: some casinos truncate at one or three decimals, and some cap the multiplier at a maximum win, which reduces the tail survival and inflates the apparent edge. The house-edge parameter may also differ from the value you assumed — read the game rules, not the promo page. Finally, replays can fail if the operator rehashes a master seed per session; a mixed sample across sessions is then compared against the wrong reference curve.

If the deviation survives those controls, extend the sample by another 10,000 rounds before writing to the casino. A statistical audit has the same logical structure as all verification-first review: the mismatch is evidence, not proof, and the resolution depends on the casino’s seed commitment for the contested rounds. Existing audit threads where players post their sample sizes and results can be found in our casino reviews. What a verified curve does establish is that the house edge is what the rules declare it to be, and that a player’s expected loss per unit staked at any multiplier is that edge, nothing worse.

FAQ

How many rounds do I need to audit a crash game?

At least 10,000 is a defensible starting point. At m = 2.00 the survival fraction has a standard error of about 0.5 percentage points, so a persistent 1-percentage-point shift appears at roughly two standard errors — enough to justify extending the sample. Tail thresholds above 10x contain only about a tenth of the sample, so either accept the larger uncertainty there or collect 50,000-plus rounds.

Why does the observed curve never match the table exactly?

Three sources of divergence are normal: flooring to two decimals shifts mass near whole-number boundaries; hash reductions that use a modulo of a smaller integer create tiny non-uniformity in h; and the house-edge parameter may be applied on profit rather than inside the multiplier. Always compare against the operator’s exact published formula, using standard-error bands for the sample size, not against the idealized 1/m curve.

Does a matching curve prove the game is fair?

It proves the noise source — the derived integer h — behaved according to the replacement rule for the sampled rounds. Fairness in the full sense requires the casino to commit to the seed chain before the rounds were played and to let you replay each round from that committed chain. A curve match without chain verification is a statistical fingerprint, not a proof; a chain mismatch with a matching curve is still a broken provable-fair guarantee.