Bankroll & variance: what the swings really are

The exact number

On 9/6 Jacks or Better with optimal play, the variance of a single hand’s return is exactly 19.51 (in squared bets — the same enumeration that reproduces the published 99.5439% return). That’s a standard deviation of 4.42 bets per hand: on a $1.25 max-bet quarter machine, one hand’s “typical” deviation is about $5.50 — against an expected loss of six-tenths of a cent. Per hand, variance outweighs edge by three orders of magnitude. Skill sets the trend; the deck sets the day.

What a session actually swings

Deviations grow with the square root of hands played. Expected loss grows linearly — but from a very small number (0.456% of each bet on 9/6 JoB). Quarter machine, $1.25 per hand, optimal play:

The planning rule that falls out: bring at least expected loss + 2σ for the length of play you intend, and treat that as a floor — mid-session drawdowns run deeper than endpoints. For an evening on quarters that’s roughly $500; halve the denomination and every number halves.

The grind between royals

Here is the psychologically important decomposition: exactly 2.0% of the game’s return rides on the royal flush, which arrives once per 40,391 hands on average. Between royals, you are playing a 97.5% game — a steady, deliberate grind that the royal repays in one spike. About 37% of full cycles pass with no royal at all. If your bankroll or your patience can’t carry the grind phase, drop a denomination; the math is identical, only the dollars shrink.

Higher-variance games need bigger rolls

Variance is a property of the paytable. Bonus-style games push more of their return into rare quads — 10/7 Double Bonus pays 160-per-coin for aces — which raises the return and the swings:

Same enumeration, same exactness: Double Bonus swings 45% harder in variance terms. An evening of 2,000 max-bet quarter hands on 10/7 DB has an expected result of plus $4.30 — and a ±1σ band of $297. That’s the honest price of the over-100% game: positive EV you can easily lose at for weeks. Start on Jacks or Better; move up when both your chart and your roll are ready.

The one variable you control

You can’t reduce variance by playing differently — correct play maximizes EV, and the swings come with it. What you control is the leak: every misplay adds a fixed, permanent cost on top of luck. We measured that cost exactly, and the trainer measures yours.

Related: the exact odds of every hand · what EV means · picking the right paytable. 18+, training and math only — never play with money you can’t lose.