Research: EuroMillions Simulation — Hot, Cold & Independence

Generated: 2025-12-15T20:30:55.662Z · Lottery: EuroMillions (synthetic; main+bonus rules)

Pre-registration

Primary hypothesis: Under independence, conditioning on a lagging first block does not make the target appear sooner. (Δ = 0).
Primary metric: Primary outcome = ΔmeanWaitToFirstHit (RMST@200) for lagging − control; one-sided direction = lagging sooner (Δ<0).
Analysis plan: We report analytic 95% CIs for ΔRMST@200 (primary), plus secondary ΔmeanCount@10 and a bounded ΔpHitWithin(X) using a scenario-specific X chosen to avoid ceiling/floor (baseline near ~0.7). We also show a robustness grid over (cutoff, window, alpha).

Design

Mode: target
Rules: main 5 of 50
Seed: 1; Future draws: 200; Grid trials per cell: 2000
Primary horizon (RMST): 200; Secondary meanCount window: 10
Grid: cutoff ∈ {50, 100, 200}, windows ∈ {5, 10, 20, 50, 100}, α ∈ {0.01, 0.05}
UX pHitWithin window: 11
Target mode: random

Primary result (pre-registered)

Mode: target; cutoff=100; alpha=0.05; window=10; RMST horizon=200

Primary outcome: ΔmeanWaitToFirstHit (RMST@200) lagging − control

Definition: draw index of first hit in the future window (1..T), with “no hit” treated as right-censored at T; reported as RMST = E[min(Tfirst, T)].

Estimate: -1.130

95% CI: -3.677 to 1.416

One-sided p (approx): 0.192107

n: lagging=45; control=1918

Note: p-value is one-sided because direction (lagging sooner ⇒ Δ<0) is pre-registered; CIs are two-sided 95%.

Censoring rate: lagging=0.000000; control=0.000000

Secondary outcome: ΔmeanCount in next 10 draws (lagging − control)

Estimate: 0.016

95% CI: -0.208 to 0.240

(Linear expectation metric; not bounded by 0..1.)

Tertiary (bounded / UX metric): ΔpHitWithin(11) lagging − control

Window is chosen to avoid ceiling/floor when possible; still bounded and non-linear.

Estimate: 0.064732

95% CI: -0.062527 to 0.191990

Kaplan–Meier survival curve: probability the target has nothit yet by draw t

Curves should overlap under independence; systematic separation would indicate a real shift in time-to-first-hit.

Interpretation: Lagging numbers do not reach their first hit sooner than comparable non-lagging numbers under independence. Shaded areas represent pointwise 95% confidence intervals.

Limitations

Simulation-based
Assumes independence
Not testing real-world rigging

Data verdict (plain language)

Verdict: No evidence of a ‘due’ effect.

ΔRMST@200 = -1.130 (95% CI -3.677 to 1.416); one-sided p≈0.192107. Estimate is slightly sooner, but not reliably different from 0.
Interpretation: “Lagging sooner” means fewer expected draws until first hit (negative ΔRMST).

Power / sensitivity (approx)

Approximate minimal detectable effect (80% power; normal approximation; two-sided alpha). Approximate also because the cohort is defined by a binomial-tail conditioning event.

Primary metric baseline meanWait (RMST@200): 10.086

n: lagging=45; control=1918

MDE @80% power(|ΔmeanWait|): 3.638

Secondary baseline meanCount@10: 1.006

MDE @80% power(|ΔmeanCount|): 0.320

Interpretation: effects smaller than the MDE are hard to reliably detect with this cohort selection + sample size.

“Approximate” reflects both the normal approximation and the binomial-tail conditioning used to define cohorts.

Tertiary metric baseline: pHitWithin(11) = 0.690824 · MDE @80% power (|ΔpHit|) = 0.195153

Interpretation notes (important)

Independence null: for toy processes (coin/die) the true effect is 0 by construction; any single “significant” result can occur by chance when you scan many cells.
Ceiling/floor risk: when baseline pHitWithin is near 1 (ceiling) or near 0 (floor), ΔpHitWithin is a bounded, non-linear metric and can look more dramatic than it is.
Preferred quantity: the primary endpoint here is mean wait-to-first-hit (RMST@T), which directly targets “does it show up sooner?” and avoids the ceiling pathology of pHitWithin in high-p regimes.
Small cohorts: grid cells with very small cohort sizes (n <20) are shown for completeness but are not interpretable (often producing degenerate CIs like 0 to 0).

What these results do not show

No “compensation” mechanism: a cold streak does not create a forward advantage under independence.
No exploitable strategy: statistically significant cells (especially under ceiling/floor metrics or small n) do not imply predictability or an edge.
No jackpot implication: this does not change the combinatorial odds of a specific full ticket.
No claim about real-world fairness: real lotteries can be tested for bias separately; this report’s main claim is about conditional reasoning under the null.

Robustness grid (lagging vs control)

Each row is one (cutoff, α, window). Look for systematic drift away from 0; do not over-interpret isolated “significant” rows.
Rows with n(lag) or n(ctrl) < 20 are visually muted and are not interpretable.

mode	cutoff	alpha	window	n(lag)	n(ctrl)	Δmean	95% CI (Δmean)	ΔRMST	95% CI (ΔRMST)	ΔpHit	95% CI (ΔpHit)
target	50	0.01	5	0	1994	NA	NA	NA	NA	NA	NA
target	50	0.01	10	0	1994	NA	NA	NA	NA	NA	NA
target	50	0.01	20	0	1994	NA	NA	NA	NA	NA	NA
target	50	0.01	50	0	1994	NA	NA	NA	NA	NA	NA
target	50	0.01	100	0	1994	NA	NA	NA	NA	NA	NA
target	50	0.05	5	8	1948	0.133	-0.384 to 0.649	-0.363	-1.461 to 0.734	NA	NA
target	50	0.05	10	8	1948	0.135	-0.646 to 0.917	-0.816	-3.459 to 1.827	NA	NA
target	50	0.05	20	8	1948	-0.131	-1.073 to 0.811	-1.784	-5.704 to 2.137	NA	NA
target	50	0.05	50	8	1948	0.234	-1.143 to 1.610	-2.974	-6.906 to 0.957	NA	NA
target	50	0.05	100	8	1948	-0.188	-2.024 to 1.649	-3.017	-6.950 to 0.915	NA	NA
target	100	0.01	5	6	1990	-0.488	-0.517 to -0.459	0.908	0.845 to 0.970	NA	NA
target	100	0.01	10	6	1990	-0.339	-0.994 to 0.316	2.470	1.222 to 3.719	NA	NA
target	100	0.01	20	6	1990	0.200	-0.865 to 1.265	2.986	-0.689 to 6.661	NA	NA
target	100	0.01	50	6	1990	-0.788	-1.969 to 0.393	1.818	-1.869 to 5.504	NA	NA
target	100	0.01	100	6	1990	0.387	-1.419 to 2.192	1.775	-1.912 to 5.462	NA	NA
target	100	0.05	5	45	1918	-0.109	-0.255 to 0.038	-0.014	-0.432 to 0.403	-0.023161	-0.166507 to 0.120185
target	100	0.05	10	45	1918	0.016	-0.208 to 0.240	-0.152	-1.133 to 0.830	0.099664	-0.027690 to 0.227018
target	100	0.05	20	45	1918	0.123	-0.270 to 0.515	-0.856	-2.579 to 0.867	0.037806	-0.046666 to 0.122277
target	100	0.05	50	45	1918	-0.185	-0.808 to 0.438	-1.086	-3.631 to 1.458	0.003128	0.000629 to 0.005627
target	100	0.05	100	45	1918	-0.116	-0.932 to 0.700	-1.130	-3.677 to 1.416	0.000000	≈0.000000 (degenerate CI)
target	200	0.01	5	9	1984	-0.205	-0.534 to 0.123	0.178	-0.735 to 1.091	NA	NA
target	200	0.01	10	9	1984	-0.027	-0.828 to 0.775	0.841	-1.537 to 3.219	NA	NA
target	200	0.01	20	9	1984	-0.010	-1.044 to 1.025	1.335	-3.105 to 5.776	NA	NA
target	200	0.01	50	9	1984	-0.163	-2.054 to 1.729	0.385	-4.319 to 5.089	NA	NA
target	200	0.01	100	9	1984	0.554	-1.805 to 2.914	0.306	-4.399 to 5.012	NA	NA
target	200	0.05	5	39	1927	-0.158	-0.345 to 0.030	0.243	-0.163 to 0.649	-0.103096	-0.252695 to 0.046502
target	200	0.05	10	39	1927	-0.339	-0.641 to -0.038	1.076	-0.021 to 2.174	-0.223677	-0.380739 to -0.066615
target	200	0.05	20	39	1927	-0.349	-0.756 to 0.059	2.810	0.542 to 5.077	-0.032933	-0.147103 to 0.081237
target	200	0.05	50	39	1927	-0.309	-1.031 to 0.413	2.774	-0.057 to 5.604	0.005189	0.001981 to 0.008397
target	200	0.05	100	39	1927	0.125	-0.674 to 0.925	2.693	-0.141 to 5.527	0.000000	≈0.000000 (degenerate CI)

How to Cite This Page

Lucky Picks. “Research: EuroMillions Simulation — Hot, Cold & Independence.”
https://luckypicks.io/research/independence-and-lagging-numbers/euromillions-simulation/
(Accessed December 2025)

For a non-technical explanation of what these results mean for players, see our Hot & Cold Lottery Numbers guide.