Why casinos are always in the black - a mathematical explanation

Neon and fanfare aside, a casino is a mathematical system with positive profit expectations at each round. Players can win short-term (and sometimes very big), but over long distances the average score of all players tends to a pre-programmed advantage at home. Below is how it works.

1) Basic formula: wait, edge and RTP

Expected value (EV) of the bet:

[
\ text {EV} =\sum p_i\cdot\text {payout} _ i -\text {bid}
]

If (\text {EV} <0) is for the player, then (\text {EV}> 0) is for the home.

House Edge (home advantage) - the average stake expected to remain at the casino.

Example: edge = 2% means that at a distance from each 100 € turnover the house "holds" ≈ 2 €.

RTP (Return to Player) - "return to the player" as a percentage of turnover.

[
\text{Edge} = 1 - \text{RTP}
]

Example: RTP 96% → edge 4%.

💡 Important distinction: RNG determines the fair chance of a particular outcome; RTP/edge sets the "payout math." RNG does not "twist" the expectation - it is laid down in advance by the rules of the game.

2) Where home advantage comes from: examples

European roulette (1 nil). Bet on Red: 18/37 winning chance, 1-1 payout.

[
\text{EV} = \frac{18}{37}\cdot 1 - \frac{19}{37}\cdot 1 = -\frac{1}{37}\approx -2{,}70%
]

This is the edge of the house ≈ 2.70%.

American Roulette (0 and 00). More "zeros" → higher than edge ≈ 5.26%.

Slots. Mathematics is spelled out in the tables of probabilities and payments: popular RTP values ~ 95-97% (edge 3-5%), but are also found below.

Blackjack. With a basic strategy and good rules, the edge of the house can fall ≈ to 0.3-0.7%. Change the rules (6:5 per blackjack, fewer splits, CSM mixer) - and the edge grows.

Sports betting in prematch. The margin is included in the "fork" of coefficients (in the sum of probabilities in excess of 100%).

3) Why a house wins "on average" rather than "every time": variance and the law of large numbers

Variance (volatility) creates a "swing" of results: short series can give the player a plus.

The law of large numbers: when the number of rounds grows (n), the average result tends to wait; the standard deviation of the mean falls as (1/\sqrt {n}).

Example: with edge 2% and player turnover 10,000 €, the expected loss ≈ 200 €, but the actual result can "walk" around this point. The more we play and the faster we spin the rounds, the closer the actual hold is to the theory.

4) Game speed - invisible edge multiplier

Even a small home advantage is scaled by the number of rounds per hour:

Roulette: ~ 40 spins/hour × 2.7% edge.
Blackjack: ~ 60-80 distributions/hour × ~ 0.5-1.5% (according to the rules).
Slots: 400-600 spins/hour × 3-6% (often) → the fastest "multiplier" of mathematics.

Total: the faster the pace, the higher the expected revenue of the house per unit of time (and the faster the player feels the variance).

5) Table limits, minimals and rule design

Up and down limits. Minimum rates accelerate turnover, and maximum rates do not allow "catch-up" strategies to endlessly "overlap" losses.

Payout rules. One change (for example, blackjack 6:5 instead of 3:2) shifts the expectation by dozens of percent beeps - at a distance these are huge amounts.

Side-bats and jackpots. Bright and rare winnings tend to be bought by a higher edge compared to the base game.

6) Interface psychology: "almost-winning" and rituals

Displaying the result (animation, "almost coincided," sounds) does not change the probability, but increases the duration of the session and turnover. The longer and faster the game, the closer the actual result is to the edge of the house.

7) Comps, cashback and bonuses: playing with net edge

Comp/cashback partially returns the player a share of the turnover (for example, 0.1-0.5%). In fact, this reduces the net edge, but rarely makes it negative for the home.

Bonuses with a vager prolong the game, and not "give profit": the real cost of the bonus is usually less than the edge included in the turnover of the game.

8) Managing the risk of the casino itself

Dom is also a player, only with a plus expectation, but with the risk of volatility.

Diversification. Many tables/games/providers → smoothing variability.

Payout limits and maximum ratios. Reduction of tail risk in jackpot scenarios.

Target by hold%. Operators plan the expected "held" percentage of revenue (hold) by vertical and monitor deviations.

Anti-advantage-play. Technical procedures (mixers, rules, account/angle protection in rates), pattern analytics, limits.

9) Rare exceptions: where a player can have a plus

Blackjack with card counting on "soft" rules and without CSM (with the right tactics and bankroll).

Video poker with the "correct" paytable (rarely, but sometimes).

Sports arbitration/value betting with a real discrepancy between "price" and probability.

Errors/promotional holes (short-lived and close quickly).

Everywhere the key: strict methodology, discipline, bank, low volume limit, constant counter-management by the operator. For a mass player, these windows are usually inaccessible or short-lived.

10) Why "big winnings" don't clash with home pluses

The house wins on average in the betting mass. An individual player or session can go into a big plus, because the dispersion of the "thick tail." But the total turnover of many players at a distance returns the system to a given edge. That is why casinos willingly advertise large winnings: they do not break mathematics, but confirm the "dream promise," increasing turnover.

Calculator on a napkin: how much an hour of play "costs"

Let the rate be 2 € per spin, 500 spins/hour, RTP 96% (edge 4%).

Expected cost per hour for player:

[
2 \times 500 \times 4% = 40,€
]

Over a long distance, this is the benchmark for "entertainment prices per hour." The actual result "walks," but around this point.

Practice for the player: how to reduce the "price of an hour"

1. Choose games with higher RTP and no "expensive" side bets.

2. Slow down. Fewer rounds per hour - lower edge multiplication.

3. Play low volatility if you want to stay "in the game" longer.

4. Use computers/cashback as a net edge reduction, but not as a "profit source."

5. Set time and budget limits before the session starts.

Casinos are "always in the black," not because they "tweak," but because the rules of gaming set a positive expectation at home. Added to this are game speed, limits, interface design and operator risk management. The player can win short-term - this is the thrill of variance - but over a long distance, the sum of all bets inevitably reduces the results to edge. A deliberate strategy is to understand this math and manage pace, game choices and limits, turning excitement into controlled entertainment.