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.