Research: Dice Toss Simulation — Hot, Cold & Independence

Generated: 2025-12-15T20:18:04.166Z · Lottery: Dice Toss (1..6)

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 1 of 6
  • 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: 7
  • 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: -1.099

95% CI: -2.463 to 0.266

One-sided p (approx): 0.057309

n: lagging=47; control=1908

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: -0.083

95% CI: -0.411 to 0.246

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

Tertiary (bounded / UX metric): ΔpHitWithin(7) lagging − control

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

Estimate: 0.079999

95% CI: -0.034249 to 0.194248

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 = -1.099 (95% CI -2.463 to 0.266); one-sided p≈0.057309. 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): 5.843

n: lagging=47; control=1908

MDE @80% power(|ΔmeanWait|): 1.950

Secondary baseline meanCount@10: 1.700

MDE @80% power(|ΔmeanCount|): 0.469

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(7) = 0.728512 · MDE @80% power (|ΔpHit|) = 0.183860

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 2 1996 -0.325 -1.305 to 0.656 -0.111 -3.052 to 2.830 NA NA
target 50 0.01 10 2 1996 -1.147 -2.128 to -0.165 0.890 -6.951 to 8.732 NA NA
target 50 0.01 20 2 1996 -0.797 -1.779 to 0.186 0.433 -8.390 to 9.256 NA NA
target 50 0.01 50 2 1996 -1.211 -1.327 to -1.095 0.250 -8.574 to 9.074 NA NA
target 50 0.01 100 2 1996 -2.020 -3.012 to -1.027 0.250 -8.574 to 9.074 NA NA
target 50 0.05 5 43 1930 -0.181 -0.451 to 0.090 0.286 -0.173 to 0.744 -0.157621 -0.307657 to -0.007585
target 50 0.05 10 43 1930 -0.303 -0.664 to 0.057 0.785 -0.231 to 1.801 -0.034173 -0.156944 to 0.088598
target 50 0.05 20 43 1930 -0.279 -0.832 to 0.273 1.018 -0.619 to 2.656 -0.015942 -0.079353 to 0.047470
target 50 0.05 50 43 1930 -0.021 -0.863 to 0.820 1.138 -0.778 to 3.054 0.000000 ≈0.000000 (degenerate CI)
target 50 0.05 100 43 1930 -0.469 -1.721 to 0.783 1.138 -0.778 to 3.054 0.000000 ≈0.000000 (degenerate CI)
target 100 0.01 5 8 1985 -0.111 -0.602 to 0.380 0.706 -0.014 to 1.427 NA NA
target 100 0.01 10 8 1985 -0.576 -1.023 to -0.129 0.578 -1.316 to 2.472 NA NA
target 100 0.01 20 8 1985 0.174 -0.657 to 1.005 -0.079 -2.151 to 1.992 NA NA
target 100 0.01 50 8 1985 0.957 -1.198 to 3.112 -0.191 -2.264 to 1.882 NA NA
target 100 0.01 100 8 1985 0.909 -2.568 to 4.386 -0.191 -2.264 to 1.882 NA NA
target 100 0.05 5 47 1908 0.100 -0.118 to 0.318 -0.451 -0.942 to 0.039 0.131473 0.004910 to 0.258036
target 100 0.05 10 47 1908 -0.083 -0.411 to 0.246 -0.863 -1.784 to 0.058 0.018043 -0.078668 to 0.114753
target 100 0.05 20 47 1908 0.300 -0.154 to 0.754 -0.985 -2.347 to 0.377 0.025681 0.018584 to 0.032779
target 100 0.05 50 47 1908 0.255 -0.495 to 1.006 -1.099 -2.463 to 0.266 0.000000 ≈0.000000 (degenerate CI)
target 100 0.05 100 47 1908 -0.680 -1.783 to 0.423 -1.099 -2.463 to 0.266 0.000000 ≈0.000000 (degenerate CI)
target 200 0.01 5 4 1986 -0.337 -1.318 to 0.644 0.373 -1.588 to 2.334 NA NA
target 200 0.01 10 4 1986 0.325 -1.062 to 1.712 1.447 -2.352 to 5.245 NA NA
target 200 0.01 20 4 1986 0.376 -1.096 to 1.848 0.686 -3.115 to 4.487 NA NA
target 200 0.01 50 4 1986 1.159 -1.437 to 3.754 0.574 -3.229 to 4.376 NA NA
target 200 0.01 100 4 1986 1.242 -0.725 to 3.208 0.574 -3.229 to 4.376 NA NA
target 200 0.05 5 35 1935 0.021 -0.273 to 0.315 0.267 -0.251 to 0.785 -0.037874 -0.203260 to 0.127512
target 200 0.05 10 35 1935 0.185 -0.259 to 0.630 0.383 -0.694 to 1.459 -0.009671 -0.135606 to 0.116263
target 200 0.05 20 35 1935 0.437 -0.157 to 1.031 0.591 -1.108 to 2.290 0.021189 0.014772 to 0.027605
target 200 0.05 50 35 1935 0.648 -0.370 to 1.667 0.476 -1.226 to 2.178 0.000000 ≈0.000000 (degenerate CI)
target 200 0.05 100 35 1935 0.753 -0.464 to 1.970 0.476 -1.226 to 2.178 0.000000 ≈0.000000 (degenerate CI)

How to Cite This Page

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

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