Casino House Edge and Bankroll Maths: A Practical Guide
Most gambling advice is about feel: when a table is hot, when a machine is due. None of it survives the arithmetic. This guide uses the arithmetic instead — the house edge of each common game, what a session costs on average, how large the swings around that average are, and what betting systems can and cannot change. Every example states its assumptions so the numbers can be checked.
The Formula Behind Every Session
Two numbers decide what a session costs on average. The house edge is the share of every stake a game keeps in the long run. Turnover is the total of every bet you place — not what you deposit, because each bet is charged the edge separately.
Expected loss = turnover × house edge
Example assumptions: €100 deposit, 800 spins, slot at 96% RTP (4% edge).
- At €1 a spin: turnover €800, expected loss €32.
- At €5 a spin: turnover €4,000, expected loss €160 — more than the deposit, which in practice means the deposit is likely to run out before 800 spins.
Stake size and number of bets, not deposit size, drive the cost. Winnings that are played again also count as turnover, which is why turnover is usually several times larger than the money paid in.
House Edge by Game
| Game and bet | House edge | Expected cost per €1,000 staked |
|---|---|---|
| Blackjack, basic strategy, common rules | about 0.5% | about €5 |
| Baccarat, banker bet (5% commission) | 1.06% | €10.60 |
| Baccarat, player bet | 1.24% | €12.40 |
| European roulette (single zero) | 2.70% | €27 |
| Slots (RTP 92–98%) | 2–8% | €20–€80 |
| American roulette (double zero) | 5.26% | €52.60 |
| Baccarat, tie bet (pays 8:1) | 14.4% | €144 |
Where the figures come from: on a single-zero wheel a straight-up bet wins 1 time in 37 and pays 35 to 1, so the expected return per €1 is 36 ÷ 37 = €0.973, an edge of 1 ÷ 37 = 2.70%. A double-zero wheel has 38 pockets and the same payouts, giving 2 ÷ 38 = 5.26%. Choosing the single-zero wheel halves the cost without any skill at all. For slots, the RTP in the game's information panel is the figure to read; many games exist in several RTP versions and the casino picks which one it runs.
How Big the Swings Are
The house edge is an average. What you actually experience in a session is the average plus a swing, and the swing is usually far larger.
The size of the swing is measured by the standard deviation. For an even-money roulette bet it is almost exactly the stake. Over many bets it grows with the square root of the number of bets, while the expected loss grows in a straight line.
Example assumptions: €10 on red, European roulette, 100 spins.
- Expected loss: 100 × €10 × 2.70% = €27.
- Standard deviation: €10 × √100 = €100.
- About two sessions in three end between −€127 and +€73; about one in six ends more than €127 down.
That is why short sessions feel random and long ones feel expensive: over 100 spins the swing (€100) dwarfs the edge (€27), but over 10,000 spins the expected loss (€2,700) is far larger than the swing (€1,000). Slots behave the same way with bigger swings; high-volatility slots have standard deviations many times the stake, though providers rarely publish the figure.
Losing Streaks Are Normal
A red/black bet on a single-zero wheel loses with probability 19 ÷ 37, about 51.4%. Long runs of losses look unusual but are routine over a session.
| Spins played | Chance of at least one run of 6 losses | Run of 8 losses | Run of 10 losses |
|---|---|---|---|
| 100 | 60% | 20% | 6% |
| 200 | 84% | 37% | 11% |
| 500 | 99% | 70% | 26% |
| 1,000 | almost certain | 91% | 46% |
Calculated exactly for independent even-money bets with a 19/37 chance of losing each spin. The table is the reason the Martingale fails, below.
What Betting Systems Change, and What They Do Not
Martingale, D'Alembert, Fibonacci and Labouchère change the size of the next bet based on the last result. None changes the expected loss, because every bet in the sequence carries the same edge: expected loss is still turnover × edge, and a progression usually increases turnover.
Example assumptions: Martingale on red starting at €5, doubling after each loss.
After 8 losses in a row you have lost €5 + €10 + … + €640 = €1,275, and the next bet is €1,280. Many tables cap bets well below that, and few bankrolls cover it. The table above shows a run of 8 losses has about a 37% chance of appearing within 200 spins. Each completed Martingale sequence wins only the starting €5, so a single long run wipes out hundreds of small wins. The system changes the shape of results — frequent small wins, rare large losses — not their average.
Bankroll Rules That Actually Do Something
Bankroll management does not change the expected result of any bet. It changes how likely you are to run out before the session ends, and how much is exposed in a bad hour.
- Set the session budget before you open the site, and treat it as spent once deposited. A budget decided mid-session is not a budget.
- Keep each stake at 1–2% of the bankroll. On €500 that is €5–€10. At 10% per bet, ordinary variance can end the session within minutes.
- Count turnover, not deposits. Turnover × edge predicts what the session costs.
- Use account limits. Deposit, loss and session limits set on a calm day work better than decisions made while losing. Check what the casino offers before you deposit; not every operator provides the same tools.
- Pick the game before the bet. Moving from a 5.26% wheel to a 2.70% one, or from a 94% slot to a 97% one, cuts the cost more than any betting pattern can.
Choosing Games on Something Other Than Instinct
Two published figures describe a slot. RTP is the long-run return, the mirror of the house edge: 96% RTP means a 4% edge. Volatility describes how that return arrives — low volatility pays small amounts often, high volatility pays rarely and large.
Match volatility to the bankroll. A high-volatility slot can empty a small bankroll long before its RTP means anything, because RTP is an average over millions of spins and a session is a few hundred. For a small bankroll, a lower-volatility game at a similar RTP gives more playing time for the same expected cost.
When to Stop, and Where to Get Help
Gambling is paid entertainment with a known average price. It stops being that when you chase losses, bet more than you planned, borrow to play or hide how much you play. If any of that sounds familiar, use the casino's self-exclusion or cooling-off tools and talk to someone. In the UK the National Gambling Helpline (0808 8020 133) is free and open around the clock; internationally, Gambling Therapy (gamblingtherapy.org) offers free support in several languages.
Practical Questions
Does stopping while ahead improve my results?
It does not change what each bet is worth, because every bet carries the same edge whenever it is placed. What it does is reduce total turnover, and since expected loss is turnover times edge, betting less does lower the expected cost — a real effect, just not the one usually claimed.
Is there a best time to play?
No. Online games draw each result from a random number generator that holds no record of previous rounds, so the hour, the day and how long a game has gone without paying are irrelevant to the next result. Live tables use real cards and wheels, which have no memory either.
Which casino game has the lowest house edge?
Blackjack played with correct basic strategy, at about 0.5% under common rules. Baccarat on the banker bet is about 1.06% and single-zero roulette 2.70%. The gap between the best and worst common bets is more than tenfold, which makes game choice the largest factor a player controls.
How much should I bet per spin?
One to two percent of the money set aside: €5 to €10 on a €500 bankroll. This does not improve the odds of any spin. It decides how many spins you get before variance ends the session.