Published on April 17, 2026 • 5 min read • By James Hartley
Mathematics does something real at a casino, but not what the «beat the odds» articles promise. It does not turn a negative game positive. What it does is tell you exactly what each game costs per round, which games cost the least, and how wide the swings around that cost will be. Here is the honest version, starting with the formula that gets misused most.
The Kelly criterion says «do not bet»
Kelly answers one question: given an edge, what fraction of a bankroll maximises long-run growth? The formula for an even-money bet is f = (bp − q) / b, where b is the payout odds, p the probability of winning and q the probability of losing.
Put a European roulette bet on red into it. There are 37 pockets, so p = 18/37, q = 19/37 and b = 1. That gives f = (18/37 − 19/37) / 1 = −1/37. The fraction is negative, which in Kelly's terms means the correct stake is zero — the bet is one you would want to be on the other side of. Kelly is not a staking plan for casino games. It is a test that casino games fail.
The same arithmetic runs through every game on the floor. Expected value per unit staked is simply minus the house edge, and no sizing rule changes its sign.
What the numbers actually are
| Game | House edge | Expected cost of 200 rounds at €2 |
|---|---|---|
| Blackjack, basic strategy | ≈0.5% (rule-dependent) | ≈€2 |
| Video poker, 9/6 Jacks or Better, optimal play | 0.46% | ≈€1.84 |
| Baccarat, banker | 1.06% | ≈€4.24 |
| Roulette, single zero | 2.70% | ≈€10.80 |
| Roulette, double zero | 5.26% | ≈€21 |
| Slots at 96% published return | 4% | ≈€16 |
| Baccarat, tie bet | ≈14.4% | ≈€57.60 |
That table is the whole of «strategy» for most players. Choosing the single-zero wheel over the double-zero one saves 2.56 percentage points; choosing the banker bet over the tie saves about thirteen. Neither requires practice — only reading the table or the game's information panel before the first bet.
Where a decision genuinely changes the number
Two games reward study, and in both cases the study means memorising a table rather than reading the game. Blackjack basic strategy is a grid of your hand against the dealer's card, and following it exactly is what produces the half-percent figure. Video poker goes further: because the deck and the paytable are both known, the best hold for every hand can be computed exactly — which is why the same title pays 99.54% as 9/6 Jacks or Better and 95.00% as the 6/5 version.
Everywhere else the only decision that counts is which game and which stake. Slots, roulette and baccarat offer no in-round choice that moves the return.
Betting systems and the table limit
The Martingale is the system that arithmetic dismantles fastest. Doubling after every loss from a €10 base means the eighth consecutive loss needs €1,280 on the table and €2,550 risked in total — to recover €10. On a single-zero wheel the chance of eight losing even-money spins in a row is (19/37)8 ≈ 0.48%, about one run in 207, which over a few hundred spins is an expected event rather than a freak one. The table maximum usually stops the doubling before the bankroll does.
D'Alembert, Fibonacci and Labouchère are gentler in shape and identical in principle: they redistribute variance and leave the edge untouched. A system that wins often and loses rarely and hugely is a distribution, not an advantage.
Variance is the part worth understanding
Expected loss is the average, not the experience. A 96% slot returns 96% over millions of spins; over 200 spins it can return nothing or several times the stake, because the payout distribution is dominated by rare large hits. That is why «I was up 300% last night» and «the maths says you lose 4%» are both true statements about the same game.
The practical use of that fact is sizing rather than prediction. Decide the amount and the number of rounds before starting, treat stake × rounds × house edge as the price of the session, and expect the actual result to sit somewhere either side of it. For the most complete public reference on game mathematics, the tables at Wizard of Odds are the standard.
Advantage play, and why it is not a strategy here
Positive-expectation gambling does exist: card counting in blackjack against a shallow shoe, full-pay Deuces Wild at over 100% with perfect play, sports betting where you price better than the bookmaker. All of it requires conditions casinos work to remove — continuous shuffling machines, trimmed paytables, limited accounts. None of it applies to a slot lobby, and no formula creates an edge that the game does not offer.
Frequently asked questions
Can mathematics beat casino odds?
No. It can only reduce what you pay. Choosing games with a lower house edge and playing the two games with computable correct play — blackjack and video poker — brings the cost per round down; no staking formula turns a negative expectation positive.
Does the Kelly criterion work in a casino?
It works, and its answer is zero. For an even-money European roulette bet, f = (18/37 − 19/37)/1 = −1/37: a negative fraction, which in Kelly's terms means do not bet. Kelly requires an edge, and casino games are defined by not giving one.
What is the expected loss of a session?
Stake × rounds × house edge. Two hundred spins at €2 on a game with a published return of 96% costs about €16 on average. The actual result will be scattered widely around that figure, which is variance rather than luck.
Why does the Martingale feel like it works?
Because it wins small amounts often and loses catastrophically rarely. Eight consecutive losses from a €10 base require €1,280 on the next spin and €2,550 risked in total to win back €10, and such a run happens roughly once in 207 attempts on a single-zero wheel.
