The Mathematics of Crash Games

What is a crash game?

A crash game is a multiplier-based gambling format with four moving parts:

1. A multiplier starts at 1.00× and rises continuously in real time
2. At a random, predetermined point, the game "crashes" and the multiplier freezes
3. You must press cash-out before the crash to win bet × multiplier
4. If the crash comes first, the entire bet is lost

Rounds last 5–60 seconds; real implementations run 80–120 rounds per hour. The format feels like a skill game — you chose when to bail! — but as the math below shows, the choice of cash-out point has zero effect on expected value. That tension between feeling and fact is the entire psychology of the genre.

The crash point distribution

The crash point is drawn from an inverse distribution. For a game with house edge h (say 0.03 for 3%), the return to player is r = 1 − h, and:

P(multiplier ≥ m) = r / m

That's the whole model. The probability of reaching any multiplier is the RTP divided by that multiplier. For a 97% RTP game:

Note the crossover at 2.00×: the probability is just below 50%. "Double or nothing" in a crash game is slightly worse than a coin flip — that sliver is the house edge.

Generating crash points

To generate a crash point from a uniform random number R ∈ [0, 1):

crashPoint = floor(100 × r / (1 − R)) / 100

When R is near 0, the round crashes almost instantly; when R is near 1, the crash point shoots toward infinity.

The proof: every cash-out target has the same EV

This is the central lesson of Flameout. Take a $1 bet at RTP r, cashing out at multiplier m:

EV = P(win) × payout − P(lose) × bet
   = (r/m) × m − (1 − r/m) × 1
   = r − 1 + r/m − r/m
   = r − 1

The multiplier m cancels out completely. At 97% RTP the expected value is −$0.03 per dollar wagered whether you cash out at 1.10× or 500×. Which means:

• There is no optimal cash-out target
• Conservative and aggressive strategies produce the same long-run loss rate
• The only thing your target controls is variance — how wild the ride is
• Over thousands of rounds, every approach converges to the same house edge

Crash games are "strategy-proof." The game is entirely about risk tolerance, bankroll management, and the psychological illusion of control.

Same EV, wildly different rides

EV is invariant; variance is not. The standard deviation of per-round returns grows dramatically with the target:

All four lose 3¢ per dollar in the long run. Pick your poison — that's the only choice the game actually offers.

The gambler's fallacy

Crash games are unusually good at triggering the gambler's fallacy because the round history is displayed prominently. After five straight crashes below 2×, a big one feels due. It is not. Every round is an independent draw: the probability that the next round reaches 10× is exactly 9.7% (at 97% RTP) whether the last 5, 50, or 500 rounds were low or high.

The history strip exists for the same reason it exists in real crash games — it generates false pattern-recognition. Flameout shows it to you and shows you the conditional probabilities in the stats sidebar proving the patterns are noise.

Betting systems don't work

Martingale (double after every loss) is the most popular crash-game system. It does not change the expected value — it changes the distribution of outcomes: many small winning sessions punctuated by catastrophic wipeouts when a losing streak outruns the bankroll. D'Alembert, Fibonacci, and Paroli are gentler or spikier flavors of the same non-solution.

You don't have to take this on faith. The Strategy Lab runs all five systems against the identical crash sequence — 100,000 rounds in well under a second — and shows every one of them converging to the same expected loss.

House edge in context

A 99% RTP crash game has one of the lowest edges in any casino — but edge is only half the cost equation. The other half is velocity: at 100+ rounds per hour and $10 bets, a 3% edge costs about $30/hour in expected losses. Slow games with bigger edges can cost less per hour than fast games with small ones.

Provably fair: cryptography instead of trust

Crash games pioneered "provably fair" gambling — using hash commitments so players can verify every outcome was predetermined and unmanipulated, with no trust in the casino or its auditors required. It rests on one property of cryptographic hashes like SHA-256: computing a hash is instant, reversing one is computationally infeasible, and changing a single input bit changes the whole output.

H = first 52 bits of the hash
if H % 101 === 0 → crash at 1.00×   // ~0.99% instant crashes = ~1% edge
else → crashPoint = floor((100 × 2⁵² − H) / (2⁵² − H)) / 100

Limits: provable fairness guarantees outcomes weren't manipulated after commitment. It can't prove the initial seed was truly random, that the implementation is bug-free, or that the operator won't vanish. And Flameout itself doesn't implement it — no real money is at stake here, so it would be security theater. We use Math.random() live and a seeded PRNG (mulberry32) for reproducible simulations, and teach the real system instead.

A brief history