Why maths is always on the casino side

1) Key idea: Built-in advantage

Any casino game is designed so that the average long-distance player score is negative. This is fixed by two parameters:

RTP (Return to Player) - the share of bets that is returned to players on average.
House edge - the share that the casino holds on average: edge = 1 − RTP.

If the RTP slot is 96%, edge = 4%. On the back of 10,000 CU the mathematical "entertainment price" is about 400 CU. The actual result will fluctuate, but the average trend with an increase in turnover stretches to − edge × turnover.

2) The law of large numbers: why "luck" is smoothed out

At a short distance, the result can be anything - hence the story about "hit the jackpot." But:

The more rounds played, the closer the outcome is to expectation.
With a negative expectation, the result tends to the minus in proportion to the turnover.
Only the trajectory (volatility) varies: smoothly or "torn" series, but not the final trend.

3) Dispersion: why it feels "gives, then eats"

Variance (volatility) describes the spread around the mean:

Low variance - frequent small wins, smooth curve.
High dispersion - long "empty" series and rare major hits.

Important: variance does not change the average minus. It only makes the path to it more uneven, which makes it easier for the player to attribute a pattern to chance.

4) Independence of outcomes and myths of thought

The RNG has no memory. Previous backs do not affect the next.

Gambler's fallacy: after 10 misses, the chance of winning does not grow "in debt."

Cluster illusion: randomness forms series; these are not "recoil settings."

5) Why betting progressions don't "break" the math

Martingale and its variations promise to "overlap" losses, but:

At edge <0, the expectation of each bet remains negative.
There are table limits and the final bankroll: sooner or later the series will bring to a bet that cannot be bet.
Rare catastrophic breakdowns eat up many small pluses → the average result again goes to the edge − × turnover.

6) Bookmaker's margin - same edge in sports betting

The coefficients include a margin: the sum of the market implide probabilities> 100%. Without your own accurate forecast, you pay that margin over the distance.

Rule: if your probability score 'p' does not give 'k· p> 1' (in decimal format), the rate is mathematically unprofitable.

7) Bonuses and vager: hidden "tax" edge

The bonus seems free, but the wagering creates a huge turnover:

Wagering cost ≈ Bonus × Wager × edge (admitted games).
While 'Cost> Bonus', net expectation is negative.
Even with favorable arithmetic, there remains a risk of not "surviving" to the end due to dispersion (the bankroll ends before the conditions are met).

8) Game builder: where casino profits are hidden

Roulette: European ~ 2. 70% edge; American ~ 5. 26% - "more expensive."

Blackjack: With the basic strategy, edge can be <1%, but player errors quickly return a comfortable plus to the casino.

Baccarat: a bet on a banker is usually "cheaper," on a draw - "road."

Slots: a wide range of RTP, many versions of the same game - the casino chooses the configuration.

Side-beta and exotic: as a rule, with a higher edge - attractive in emotions, unprofitable in mathematics.

9) Rare exceptions and why they don't overturn the rule

Positive EV is possible in strictly limited scenarios: an ideal strategy on rare profitable pay tables, some forms of video poker, multi-pass promos with a real "overlay," professional models in sports.

But such situations require accurate calculations, discipline, data, often quickly closed by the market/rules and accompanied by countermeasures (limits, exclusions from stocks, RTP/margin changes). For most players, access to sustained EV> 0 is virtually non-existent.

10) Practical conclusions for the player

1. Consider the "price of entertainment." Total ≈ × edge −.

2. Choose "cheap" games/bets. European roulette is better than American roulette; basic strategy in blackjack is mandatory; avoid expensive side-bats.

3. Consider volatility. The higher it is, the lower the percentage of the bankroll rate (0. 25-1% vs. 1-2% for low).

4. Limit turnover and time. The longer the session, the closer the result is to casino math.

5. Check the actual RTP/rules here. The same slot can be 96/94/92%.

6. Bonuses - through a calculator. Wager Tax = Wager × Bonus × edge; compare with bonus size and risk of ruin.

7. Don't rely on progressions. This is a turnover accelerator, which means a mathematical minus accelerator.

11) Short pre-game checklist

Do I know the edge/RTP of the selected game?

I understand its volatility and chose the rate as% of the bankroll?

Put stop loss/break profit and time limit?

For the bonus, did × Wager × edge count the Bonus and appreciate the chance to "live"?

Ready to accept that a short distance can give any outcome, but a long one pulls to − edge × turn?

The casino wins not by "magic," but by arithmetic: a fixed edge + law of large numbers. Dispersion masks this in the short term, but at a distance the average becomes your result. The better you understand these mechanics, the more consciously you choose games, the size of the bet and the duration of the session - and the more reliably you keep the excitement under control.