The maths of the crash multiplier: why the average favours the house

In the standard crash model the average multiplier has no finite value, because rare huge results keep pulling it up. The figures that matter are the median, about 1.94x at a 3% margin, and the average return per shilling staked, which is 97 cents at every cash-out target. The headline average flatters the game.

By Nairobi Ultimate Editorial (prepared with AI tools under our editorial policy) · Last updated 2026-10-06

The rule the maths starts from

This guide extends how crash games work by asking what the "average multiplier" really measures. Everything below follows from one stated rule, the standard model: the chance that a round reaches multiplier m, for any m of 1.00 or more, is (1 − h)/m. Here h is the house margin, and a share h of rounds stop at exactly 1.00x.

The margin is set to 3% (h = 0.03) in every example, which is a choice for illustration and not a figure from any real game. The rules of a real game are the ones that count. Regulation 45 of the Conduct of Gambling Operations Regulations, 2026 requires the rules to be displayed before any wager and the theoretical return to player to be publicly disclosed for each game.

Step 1: the median stop

The median is the point that half of all rounds fall short of. Set the chance of reaching it to one half and solve.

(1 − h)/m = 1/2, so m = 2 × (1 − h) = 2 × 0.97 = 1.94

Half of rounds end below 1.94x. At 2.00x the chance of reaching it is 0.97/2 = 48.5%, so 51.5% of rounds end before it. A target that looks like a fair coin toss is slightly worse than one.

Step 2: why the plain average does not exist

For a quantity that cannot be below zero, the average equals the area under its "chance of exceeding" curve. Here that curve is 1 up to 1.00x (every round reaches at least 1.00x as a figure) and (1 − h)/x beyond it.

The area from 1 to a ceiling c is (1 − h) multiplied by the natural logarithm of c. The logarithm keeps growing as c grows, so there is no upper limit to the sum. The average of the stopping point is therefore infinite in the model if the game has no ceiling.

average of the stop, capped at c = 1 + (1 − h) × ln(c)

A real game has some ceiling on its multiplier, and the capped formula shows what that does. The average depends on the ceiling and on how far out the rare results reach, not on how the typical round behaves.

What the capped average looks like

Illustrative example, standard model with a 3% margin. Raise the ceiling from 100x to 1,000x and the average climbs from 5.47x to 7.70x, while the median stays at 1.94x. A history screen that reports a mean over a few hundred rounds is mostly reporting how many rare large results happened to fall inside it.

Step 3: what a stake actually earns

A player does not receive the average multiplier. A player receives a payout of t times the stake when the round reaches the chosen target t, and nothing otherwise. The average return per shilling is the payout times its chance.

t × (1 − h)/t = 1 − h = 0.97

The target t cancels. At 1.50x, 5.00x or 100.00x the average return is KES 97 per KES 100 staked, which is the game's return to player. The house margin of KES 3 is the same everywhere along the curve, and it is the figure that regulation 20(2) expects licensees to display as house edge and average return. The expected value guide explains the same idea for any bet.

Why a high average still favours the house

An average multiplier of 5x or 7x sounds like a payout of 5x or 7x. It is not, for two reasons that follow from the steps above.

First, the average is carried by results that almost never occur. In the model a round reaches 100x with probability 0.97/100 = 0.97%, yet the stretch of results between 10x and 100x, which together occur in only 9.7% of rounds, adds 0.97 × ln(100/10) = 2.23 to the capped average. A player aiming at 100x loses the stake in 99.03% of rounds.

Second, collecting the average would need the player to know the stop in advance. Cashing out at a fixed target takes only the part of the curve below that target, and that part is priced at 1 − h. The gap between the high headline average and the 97 cents is the margin plus the value of knowledge nobody has.

Rare results in a short session

The chance of seeing at least one round that reaches m in n rounds is 1 minus the chance that none does.

chance = 1 − (1 − 0.97/m)^n

For m = 100 the chance of reaching it in one round is 0.97%, so in 100 rounds the chance of at least one is 1 − 0.9903^100, which is about 62%. Seeing a 100x result on a history bar is thus ordinary over many rounds. It says nothing about the next round, because rounds are independent, which is the point of the gambler's fallacy guide.

Spread is a separate question

Because every target returns the same 97 cents, the choice of target changes only the pattern of outcomes. Low targets lose a little often and high targets lose a lot rarely, as set out in the guide to volatility. The mean does not tell you which pattern you will meet in a session.

The same arithmetic applies to the money that passes through the game. If a total of KES 20,000 is staked across many rounds, the average cost at a 3% margin is KES 600, whichever targets were used. A session limit set before play is a limit on that total.

What these figures do not tell you

The model assumes independent rounds and a margin taken as a share of rounds ending at 1.00x. Real games may build the margin differently, add rounding of the displayed multiplier or cap the maximum. Gambling is addictive, and the maths describes averages, not the result of any session. Play responsibly, and use the limits that licensees are required to offer.