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How probability theory works in gambling
1) The bottom line: what keeps the mathematics of excitement
Randomness → outcomes are unpredictable individually.
Probability (P) → proportion of favorable outcomes over a long distance.
Expectation (EV) → the average long distance bet.
Home Advantage (House Edge) → (-EV) player as% of bet.
Variance/volatility → the swing of the result around the EV.
The law of large numbers → the average result tends to EV with an increase in the number of bets, but the path is "zigzag."
2) Basic formulas (no pain)
Waiting for one bid:[
EV =\sum _ i P_i\cdot\text {payout} _ i -\text {bid}
](equivalent to: sum of "odds × net gain").
Home Advantage:[
\ text {HE} = -\frac {EV} {\text {bid} }\times 100%
][
P (\text {series} k) = p ^ k
][
P=\binom{n}{k}p^k(1-p)^{n-k}
][
\ sigma _ {\text {sum}} =\sqrt {n }\cdot\sigma _ {\text {bets}}
](important for bankroll "wobbles").
3) Tape measure on a napkin: why minus 2.7 %/5.26%
European roulette (37 pockets, one zero).
Bet on red: win + 1 with probability (18/37), lose − 1 with probability (19/37).
[
EV = \frac{18}{37}\cdot 1 + \frac{19}{37}\cdot (-1)= -\frac{1}{37}\approx -2{,}70%
]American (38 pockets, two zeros): (EV = -2/38\approx -5 {,} 26%).
Conclusion: the type of table is more important than "flair."
4) Slots: RTP and Volatility
RTP (Return to Player) - theoretical return percentage for an infinite game (for example, 96%). Then (HE\approach 4%).
Volatility: low - frequent small payouts; high - rare large (and large "drawdowns").
RTP is the length of the distance, and the player's feeling is set by the variance.
5) Blackjack, dice, betting with different EVs
Blackjack: With basic strategy, home. the advantage of the casino is often <1% (the rules affect: the number of decks, delivery from any total, etc.).
Bones (craps): different fields - different HE (pass/don't pass lines are usually softer, "exotic" bets are tougher).
Moral: The same game can have dozens of bets with different EVs.
6) Independence of outcomes and "player error"
The wheel/spin does not remember the past. 10 blacks in a row do not make red "proper."
Gambler's Fallacy: belief that nature will "equalize" frequencies in a short series. In fact, you see fluctuation, not long-term alignment.
7) The law of large numbers and "swing"
As the number of bets increases, the average profit/loss approaches (EV), but the standard deviation increases as (\sqrt {n}).
Therefore, long streams and "sick" drawdowns are possible even with a small HE.
8) Series and "almost-win": why the brain is caught
Almost winning gives the peak of emotions → it seems that the "chance has grown," although (P) is the same.
The series are explained by the binomial model: at (p = 0 {,} 5) a series of 6 consecutive victories is not uncommon on large n.
9) Risk of ruin and bankroll
Even with a "soft" HE, there is a high probability of "encountering" a drawdown that exceeds the margin.
Small rate rule: 0.5-1.0% bankroll per decision reduces the chance of rapid bankruptcy.
Kelly's criterion (theory): the optimal share for positive EV (f ^ =\frac {bp-q} {b}) (where (b) is the net profit ratio, (p) is the winning chance, (q = 1-p)).
In casino games, EV is more often negative, which means Kelly → 0 (play for fun, not as an investor).
10) "Quick Calculators"
EV lines:[
EV = P (\text {win} )\cdot\text {net payout} - P (\text {loss} )\cdot\text {bet}
]Series rating: the probability of ≥ k victories in n games is the sum of binomial tails.
Bet contribution to wagering (for bonuses): offset (\text {bet }\times\text {% contribution}).
11) Typical cognitive traps
Selective memory: remember drifts, forget long cons.
The effect is "already invested": "I will add to the round number" - irrational if the goal of entertainment has already been achieved.
Illusion of control: Changing your slot/desk doesn't change your EV if the rule is the same.
Martiingale: Geometric rate rise doesn't change (EV), but it dramatically increases risk of rapid catastrophe.
12) Practical conclusions (not advice on "deception," but common sense)
1. Know HE: European roulette is better than American roulette; "gingerbread" rates are usually worse than "base" rates.
2. Take dispersion: long drawdowns are the norm, not a "broken luck" signal.
3. The rate is small and even: reduces the risk of ruin.
4. Stop limits on time and money: probability does not "compensate" for emotions.
5. Bonuses ≠ free money: their EV depends on WR, cap, max bet and the contribution of the games.
6. Cashout - according to plan: fix part of the profit; a single skid does not "oblige" to increase the bet.
13) Mini-analysis on numbers
Example 1 - Even-Money rate in Europe. to roulette
Rate €10, (EV = -2 {,} 7%) → average long-term minus €0.27/rate.
For 1,000 spins, it is expected to − €270 on average, but in fact the spread can be from a large plus to a deep minus - this is the variance.
Example 2 - RTP slot 96%
For €1,000 spin, the wait − €40.
High volatility gives a chance of a big win and the risk of rapid drawdown; low - more "even" schedule.
14) Frequent Questions Short
Is it possible to "sit out" to a plus? The likelihood of a plus does not grow with the "fatigue" of the casino; only the risk of running into limits/bank is growing.
Series - a sign of twisting? No, the series are typical of random processes. It is necessary to evaluate honesty instrumentally (RNG audit, licenses).
Is there a "winning" watch? For games with RNG/wheel - no: the probability does not depend on the time of day.
15) Common sense checklist
- Understand my EV/HE at the selected rate
- Rate ≤ 1% bankroll, no progressions
- Loss/Time/Profit limits to withdraw in advance
- Aware of variance: drawdowns are not a "sign"
- Bonuses I take only with clear mathematics (WR, cap, contribution)
- Playing for fun, not "until lights out"
Probability theory does not "break" excitement - it puts an expectation frame: games with a fixed negative EV at a distance are taken to minus, and the "swing" of the result is explained by variance and the laws of chance. When you know HE, RTP, the independence of outcomes, the role of variance and the risk of ruin, myths disappear and a clear strategy of behavior appears: a small bet, limits, discipline and an understanding that cashout is a decision, not the mercy of luck.