Casino math: Why the house always wins

1) The main principle: built-in expectation in favor of the house

The casino earns not due to "secret algorithms," but due to the mathematical advantage in each game.

RTP (Return to Player) - the share of bets returned to long distance players.

House edge - the share that the casino holds on average: edge = 1 − RTP.

Consequence: with any turnover, the average financial result of the player tends to − edge × turnover. A short distance can give any result, but on a long one - mathematics will "catch up."

2) The law of big numbers: short-term versus distance

On a short segment, dispersion can give a large plus. However, the longer you play, the closer the actual profit/loss is to the mathematical expectation:

Total ≈ × edge −.
That is why the casino is interested in playing time and turnover, and not in the fact that you "immediately lose everything."

3) The role of volatility: the path to the average

Volatility (variance) describes the spread of results:

Low - frequent small winnings, even trajectory.
High - long empty episodes and rare major hits.
Important: volatility does not change the average minus, it only makes the road to it smooth or ragged. Because of this, myths about "hot/cold" games are born.

4) Examples of edge by game (why some are "more expensive" than others)

Roulette: European ~ 2. 70% edge, American ~ 5. 26% - the game is literally twice as "expensive."

Blackjack: with the basic edge strategy <1% (depends on the rules); errors dramatically increase the cost of the game.

Baccarat: a bet on a banker is usually cheaper (~ 1-1. 2% edge), a draw is expensive.

Bones (craps): pass/don't-pass with "odds" - one of the cheapest bets; prop rates are expensive.

Slots: more often 3-6% edge, but the range is wide; the same slot may have different RTPs in different casinos.

5) Betting margin - "house edge" in sports betting

In the coefficients, the margin is sewn up: the sum of the market odds> 100%.

Quick verification of the plus value of a single bet (decimal factor k):

EV = k· p − 1, where p is your real probability estimate.
If k· p ≤ 1, the rate is unprofitable on average: the margin "eats up" your result.

6) In-pocket formulas

edge = 1 − RTP (in fractions).

Session summary: '≈ × turnover (− edge)'.

Turnover: 'bet × number of rounds' (sum of all bets, not balance change).

Sport (single): 'EV = k· p − 1'.

The "price" of the bonus: 'Cost ≈ Bonus × Wager × edge (allowed games)'.

7) Why betting progressions don't work

Martingale and options promise to "block" the cons, but:

the expectation of each bet remains negative;
table limits and ultimate bankroll are imminent;
rare long series eat up a lot of small winnings.
The result at a distance is again equal to − edge × turnover, just the path to it is more expensive and more nervous.

8) Bonuses and cashback: where mathematics is hiding

The bonus looks "free," but the wagering creates a gigantic turnover - and with it the "tax" edge.

If'Bonus

Even with an arithmetically positive scenario, there remains a risk of not "surviving" to the end due to volatility (the bankroll will run out before the conditions are met).

Therefore, look not only at the size of the bonus, but also at the vager, the list of allowed games, the bet limit and the validity period.

9) Independence of outcomes and cognitive traps

RNG with no memory: Past spins don't affect the next one.

Player error: "after 10 cons, the chance has increased" is incorrect for independent tests.

The illusion of clusters: chance naturally creates series; this is not a "twist."

Knowing these effects saves money and nerves.

10) Rare exceptions: when EV may be> 0

There are windows where the player gets an advantage: ideal payment tables in video poker, special rules in blackjack, overlays in promo/tournaments, value-betting with a reliable model. But these situations:

rare and short-lived, require precise strategy/data/discipline, meet countermeasures (limits, rule changes, stock exclusions).
For most players, sustained EV> 0 is not available.

11) Practice: How to play mindfully

1. Choose "cheap" products: European roulette instead of American; basic strategy in blackjack; avoid expensive side-bats.

2. Rate in% of bankroll:

low volatility: 1-2%, average: 0. 5–1. 5%, high: 0. 25–1%.
3. Limits: pre-set stop loss (e.g. − 20% of session budget), break profit (+ 30% and above for high-volatility games) and time limit.
4. Sales control: remember, you pay not with a deposit, but with a percentage of the turnover.
5. Calculator Bonuses: Count 'Bonus × Vager × edge' and check game exclusions/bet limit.
6. Diary of sessions: record the turnover, the result, the depth of drawdown - this makes it easier to keep discipline.

12) Short checklist before start

Know the actual RTP/edge of the selected game in this casino?

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

Put stop loss/break profit and time limit?

For the bonus, did you count the real cost of playing and the risk of not "living"?

Ready for the fact that a short distance is accidental, and a long one draws a − turn to × edge?

The house "always wins" at a distance, because this is how the rules are arranged: a fixed edge + the law of large numbers. Volatility masks this in the short term, creating the illusion of chances to outplay the system. Consciously choosing "cheap" games, reasonable bets, strict limits and sober calculation of bonuses turn excitement into controlled entertainment - and leave mathematics on their side, not against you.