Bankroll and variance for roulette

Welcome to the lesson on roulette bankroll and variance.

I'm Annabel, and let me start with the most important statement in this lesson: the Kelly Criterion, the standard formula for optimal bet sizing used by professional gamblers and quantitative traders, gives a negative answer for roulette.

The formula says bet a fraction of your bankroll equal to the edge minus the counter-edge, divided by the payout ratio.

On European roulette red or black, the probability of winning is eighteen out of thirty-seven, approximately forty-eight point six five percent.

The probability of losing is nineteen out of thirty-seven, approximately fifty-one point three five percent.

Kelly gives: forty-eight point six five minus fifty-one point three five equals negative two point seven percent.

Kelly is telling you to be the casino, not to bet on roulette at all.

Some players discover Kelly and assume that any positive fraction of it represents a conservative, mathematically grounded approach.

A fraction of a negative number is still negative.

Multiplying negative two point seven percent by any positive fraction produces a bet that costs money.

Recreation is a valid reason to depart from Kelly's recommendation.

But departing knowingly, rather than in ignorance of what the formula actually says, is the relevant distinction.

The one case where Kelly's answer changes is French roulette with La Partage on even-money bets.

La Partage returns half your stake when zero lands.

Kelly gives exactly zero: bet nothing, break even in expectation.

La Partage halves the house edge but doesn't eliminate it.

Aspers at Westfield Stratford applies La Partage on its European tables, making it the most mathematically favourable standard roulette option at a UK land-based casino floor.

For session play, the relevant concept is path-dependent risk of ruin: not whether you're down more than a set amount at the end of the session, but whether your bankroll touched zero at any point during it.

This matters because path-dependent ruin probability is approximately double the simple end-of-session calculation.

A player who sizes their bankroll against end-of-session loss probability is underestimating actual ruin risk by roughly half.

The path-dependent ruin probability below half a percent over that session requires approximately twenty-five thousand pounds as session bankroll.

That's the fifty-times-stake rule: fifty times your stake as session bankroll produces roughly half a percent ruin probability on even-money bets over a four-hour session.

Drop to twenty times stake, ten thousand pounds, and path-dependent ruin probability climbs to approximately thirty-eight percent.

That is not a conservative position.

For straight-up bets at the same five hundred pounds per spin, the risk profile changes substantially.

The standard deviation per spin on a straight-up bet is approximately five point eight four times your stake, versus approximately one times for even-money bets.

You need approximately six times the bankroll to achieve the same ruin probability as an even-money player.

At five hundred pounds per spin on straight-up bets, a twenty-five thousand pound bankroll gives approximately sixty-three percent path-dependent ruin probability over a four-hour session.

You need around one hundred and forty thousand pounds to get that figure below three percent.

This explains why serious players who prefer straight-up bets bring very large bankrolls.

Here is the trap most recreational players walk into.

Over a short session, the variance is large relative to the expected loss, so positive outcomes are entirely plausible.

The expected loss for thirty spins at five hundred pounds per spin on even-money bets is only four hundred and five pounds.

The standard deviation is approximately two thousand seven hundred and thirty-nine pounds.

A player can easily finish five thousand pounds up without that being particularly improbable.

This creates the impression that the game is running well, or that a system is working.

Over long runs, the expected loss accumulates linearly while variance grows only as the square root of spins.

The house edge signal gets clearer over time.

This is variance doing what variance does.

Players who book those wins as evidence of sound technique and conclude their bankroll requirements are lower have made a survivorship error.

Over a three-session weekend at five hundred pounds per spin, two hundred and forty spins each: total action three hundred and sixty thousand pounds, expected loss approximately nine thousand seven hundred and thirty pounds, combined standard deviation approximately thirteen thousand four hundred and nineteen pounds.

The ninety-five percent confidence interval runs from approximately minus thirty-six thousand to plus sixteen thousand five hundred.

A player who refuses to accept the possibility of a minus thirty-five thousand weekend has not understood their own bankroll requirements.

Size the bankroll for the variance.

Know what fifty times stake actually means.

Then decide whether to sit down.