How to analyze winning streaks

"Winning streak" is consecutive successful outcomes (hits) between two failures. In fair play (independent spins), series are natural: randomness generates clusters. Competent analysis of batches helps to understand the risk profile (how often it "goes") and adjust the limits. He doesn't predict the next spin.

1) Base model: Bernoulli and batch geometry

Let each spin be an independent test with a probability of success (p) (for example, "any win" or "significant win ≥×10").

The length of the winning streak (K\ge1) before the first loss is geometrically distributed:

[
\mathbb{P}(K=k)=(1-p),p^{k-1},\quad \mathbb{E}[K]=\frac{1}{1-p},\quad \mathrm{Med}(K)\approx \left\lceil \frac{\ln 0. 5}{\ln p}\right\rceil.
]

Probability of a series of length ≥ (k): (\mathbb {P} (K\ge k) = p ^ {, k-1}).

The expected number of runs (all lengths) per (N) spin ≈ (N (1-p)).

Expected number of ≥ (k) series per (N) spins ≈ (N (1-p), p ^ {, k-1}).

💡 If "success" is a rare event (for example, ≥×10 with probability (q)), just substitute (p = q) - everything works higher for such "significant" series.

2) What exactly to measure on your logs

First, determine what constitutes success:

"any win" (HF), or
"significant" (threshold, e.g. ≥×5/×10), or
"plus spin" (payout ≥ rate).

Next, count:

1. HF (score (p)): proportion of successful spins.

2. List of winnings: (K_1,K_2,\dots) (and separately - for "significant").

3. Batch length quantiles: median, 75th, 90th percentiles.

4. Max W-streak on line (N).

5. Number of runs ≥ (k) for multiple thresholds (k) (e.g., ≥3, ≥5).

6. Losing streak statistics (L-streak) - symmetrical, this is important for stop loss on the backs.

3) Quick digit interpretation

If the observed frequencies (# {K\ge k }/#\text {series}) are close to (p ^ {k-1}), the behavior is similar to independent.

Deviations on short samples are normal. See uncertainty intervals (bootstrap by list (K_i)) and/or simulations.

Max W-streak grows logarithmically in (N): long "beautiful" series occur even with a small (p).

Mini example. Let HF (p = 0 {,} 30). Then:

(\mathbb{P}(K\ge3)=p^2=0{,}09); on (N = 1000) spins we expect (\approach N (1-p) p ^ {2 }\approach 630\times0 {,} 09\approach 57) ≥3 series. For ≥6: (p ^ {5 }\approx 0 {,} 00243) ⇒ ≈ (630\times0 {,} 00243\approx 1 {,} 5) series are rare, but not a miracle.

4) Hypothesis tests: "are the episodes too high?"

Use one or more of the following tools:

1. Comparison with geometry.

Rate (p =\widehat {HF}).

Construct theoretical (\mathbb {P} (K\ge k) = p ^ {k-1}) and compare with empirics.

Add confidence bands (bootstraps) for observed fractions.

2. Wald-Wolfowitz test (runs test).

Classify backs as success/failure.

Compare the number of "runs" with that expected at independence.

Significant deviations may indicate dependence (or just a small sample).

3. Monte Carlo under zero.

With (p) fixed, simulate thousands of sequences of length (N).

Look at the Max W-streak distribution and the number of batches ≥ (k).

Compare your observations with this distribution (p-value "too unusual or not").

💡 If "success" is selected as a rare thing (for example, ≥×10), use only spins with binarization at this threshold: 1/0.

5) Practice: how to make calculations (without a code)

1. Collect the log: back number, result (multiplier), binary flags "success," "significant success."

2. Run through the success column and form the length of the series (counter, reset to 0 if unsuccessful).

3. Calculate:

(p =) average success flag;
quantiles (K);
– Max W-streak;
frequencies (# {K\ge k}) for (k = 2.. 7).
4. Construct the theory: (p ^ {k-1}) and the expected number of series ≥ (k): (N (1-p) p ^ {k-1}).
5. Make a simulation of zero (at least 10k runs) - the distribution of Max W-streak and the number of series ≥ (k).
6. Compare and conclusion: "Within expectations "/" above expectations, but fits into confidence bands "/" suspicious - not enough data."

6) Typical traps

Selective window selection. We took a "successful" period - the series seem like magic. Use a fixed window length (for example, 1000 spin batches).

Changing success criteria on the fly. First, decide what "success" is and do not change according to the result.

Confusion of "win series" and "plus spin series." These are different binarizations (HF vs "payout ≥ rate").

Interpretation as prediction. The series describe the past pattern without reporting anything about the next back (independence).

7) How to use batches in risk management

Back limits. Knowing the quantiles of the losing series (L-streak), set the "time-out after L≥k."

Bank plan. If the median winning streak is short and "meaningful" rare, bank on "deserts."

Session length. The probability of encountering a series of ≥ (k) increases with (N). If your goal is to "catch ≥×10," evaluate (q =\mathbb {P} (\ text{≥×10 per spin}) and use (\mathbb {P} (\text {do not catch per} N) = (1-q) ^ N).

Disable Dogon. Series do not provide an advantage for increasing the rate - this is just a form of variance.

8) Mini-template for your articles/reports

Success criterion: (any win/ ≥×10/plus spin)

HF (score (p)): ...%

Quantiles of W-series lengths: median...; 75th...; 90th...

Number of batches ≥3/ ≥5/ ≥6: actual .../.../...; waiting for (N (1-p) p ^ {k-1}) .../.../...

Max W-streak: fact...; simulation range (Q5-Q95):... -...

Output: model fit/more data needed; recommendations on limits.

9) Small landmarks (to calibrate intuition)

At HF (p = 0 {,} 25): median W-series ≈ 1-2, (\mathbb {P} (K\ge5) = p ^ {4 }\approx 0 {,} 39%). On (N = 2000) spins, waiting for ≥5 series: (\approach 1500\times0 {,} 0039\approach 6).

With a rare event (q = 1%) (for example, ≥×10): the median length of the "series of significant" = 1 (rarely 2 + in a row), and the distances between such spins are large; batch analysis is more useful in terms of "pause between events" than "consecutive."

10) Analyst Short Checklist

Have I clearly fixed the criterion for success?

Is the window length and data volume sufficient (batches, more than one run)?

Compared with geometry and Monte Carlo under the same (p)?

Showed quantiles and Max W-streak with confidence bands?

Conclusions relate to risk management, not "timing" rates?

Bottom line: winning streaks are a normal form of chance. Their analysis is working with a geometric distribution and comparing observations to a null model (and/or simulation) rather than looking for a "hot clock." In gray numbers - HF, quantiles of lengths, the expected number of series and the distribution of the maximum series - you arm yourself for bank planning, session duration and limits, remaining within the framework of honest mathematics, not superstition.