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Why even the best strategy doesn't beat variance
In gambling, the outcome of a session is the sum of independent random outcomes. Each outcome has a mathematical expectation (expected return) and variance (spread). The strategy is able to redistribute risk over time (bankroll curve, frequency of "valleys" and "peaks"), but is not able to cancel the variance and, if the expectation is negative, it is not able to turn minus into plus.
1) What is variance and why it "wins"
Consider a random variable (X) - a multiplier for the spin/bet (how many times the bet returned).
Wait: (\mu =\mathbb {E} [X]) (RTP = (\mu\times100%)).
Variance: (\sigma ^ 2 =\mathrm {Var} (X)) - measures the spread of outcomes.
Over (N) independent attempts, the mean (\bar {X}) fluctuates around (\mu) with standard error[
SE=\frac{\sigma}{\sqrt{N}}.
]Even with a large (N), the spread does not disappear instantly: it falls only as (1/\sqrt {N}). Over a short distance, variance dominates the "logic" of any strategy.
2) What the strategy can and cannot
May:- change the risk profile: the length of the series of losses, the depth of drawdowns, the likelihood of rare "comebacks";
- control time (stop loss/teik profit), reducing exposure;
- select the volatility of the game for the purpose of the session (frequent little things vs rare large ones).
- change (\mu) in fair play without distortion (RTP, "house edge");
- remove variance (\sigma ^ 2);
- make independent events dependent (no "timing" will give birth to memory in RNG).
3) Why negative expectation is not defeated
If there is a house edge, then (\mu <1) (RTP <100%). Then the expected amount for (N) bets under (b):[
\ mathbb {E} [\text {win}] = N\cdot b\cdot (\mu-1) <0.
]The strategies of progressions (martingale, d'Alembert, Fibonacci) only shift the distribution: they make more short "small victories" at the cost of rare but catastrophic failures without changing (\mu).
4) "I saw how the strategy works!" - about sampling and luck
At a short distance, the noise is great:- the law of large numbers speaks only of convergence on average at a huge (N);
- the central limit theorem gives a bell around (\mu), but a width of (\propto\sigma/\sqrt {N});
- a rare large gain will easily "repaint" 500-1000 spins, creating the illusion of a "working pattern."
5) Risk of ruin and bankroll
Even with a neutral/positive expectation (for example, bonus hunting, betting advantages), variance creates a risk of ruin: the likelihood that drawdown will reach zero before the advantage is realized.
The higher the volatility and the bank's share in the rate, the higher the risk of ruin.
Stop loss limits the depth of drawdown, but does not make waiting positive. It only captures the risk.
6) Bet size and Kelly
Kelly's formula (for advantage games) maximizes the rate of capital growth by choosing a share (f ^) from the pot. But:- if the expectation is negative, (f ^ <0) ⇒ the correct bet is zero (do not play);
- with positive anticipation, Kelly reduces the risk of ruin, but does not remove the variance: the series and drawdown will remain.
7) Parsing popular strategies
Martingale: A high chance of a small plus but explosive "limit/bank wall" risk. The distribution becomes "thick-tailed" - rare huge cons.
Flat bet: the real (\mu) is seen cleaner, the variance manifests itself "honestly."
Ladders/Series Dogon: Rely on player error and "cluster illusion." The probabilities of outcomes do not change.
Stop losses/teik profits: a tool for controlling behavior and exposure time. The expectation is the same.
8) Why "perfect management" doesn't turn a minus into a plus
Any control is a filter by time (when in/out) and by position size. If (\mu <1), then the integral of your expectations in time remains negative. You can:- get a smoother curve;
- less often to meet "black swans" (at the cost of reducing the chance of rare "comebacks");
- better to experience variance psychologically.
- But the math of winning remains the same.
9) Practical conclusions for the player
1. Define the expectation. If RTP is <100% and there is no external advantage, there is no strategy that changes the waiting sign.
2. Choose volatility for the target. Want "motion" - above HF and below dispersion; if you want a chance for a "skid" - get ready for "deserts."
3. Put the size from the bank, not from emotions. High rate share = exponential increase in ruin risk.
4. Plan for drawdowns. Keep the bank's reserve under typical series: focus on the median and 75th percentile of the intervals between significant events.
5. Fix the rules before the game. Stop loss for money and backs, time out after a long L-series.
6. Measure, not "feel." Read the actual RTP, HF, interval quantiles; avoid "timings" and superstitions.
10) Mini "risk passport" template for your reviews
RTP: ...%
HF (any win): ...%
Quantiles of ≥×10 intervals: median... spins; 75th percentile...
Expected RTP spread per 1000 spins: ≈... pp (for this volatility)
Typical drawdowns (empirical): median... rates; 95th percentile...
Rate recommendations: ...% of the bank (if the goal is to keep the risk of ruin ≤...%)
Bottom line: variance is a fundamental property of randomness, not a "glitch" that can be fooled by a cunning progression. Strategies are useful for controlling risk and behavior, choosing the pace and duration of the session, but not for changing expectation and not for "defeating variance." If the expectation is negative, the only way to "beat the variance" in the long run is to reduce the exposure or not play.