Research: Coin Toss Simulation — Hot, Cold & Independence

Generated: 2025-12-15T20:17:50.557Z · Lottery: Coin Flip (Heads=1, Tails=2)

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: coin
  • Rules: main 1 of 2
  • 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: 2

Primary result (pre-registered)

Mode: coin; 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: -0.109

95% CI: -0.435 to 0.218

One-sided p (approx): 0.257545

n: lagging=61; control=1939

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: 0.040

95% CI: -0.369 to 0.449

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

Tertiary (bounded / UX metric): ΔpHitWithin(2) lagging − control

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

Estimate: 0.023715

95% CI: -0.083575 to 0.131005

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.

S(t) draws 0.5 1.0 0.0 Lagging Control

Data verdict (plain language)

Verdict: No evidence of a ‘due’ effect.

  • ΔRMST@200 = -0.109 (95% CI -0.435 to 0.218); one-sided p≈0.257545. 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): 1.994

n: lagging=61; control=1939

MDE @80% power(|ΔmeanWait|): 0.467

Secondary baseline meanCount@10: 5.009

MDE @80% power(|ΔmeanCount|): 0.584

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(2) = 0.746777 · MDE @80% power (|ΔpHit|) = 0.158331

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)
coin 50 0.01 5 9 1991 0.169 -0.398 to 0.737 NA NA NA NA
coin 50 0.01 10 9 1991 -0.360 -1.017 to 0.297 NA NA NA NA
coin 50 0.01 20 9 1991 -0.885 -1.995 to 0.224 NA NA NA NA
coin 50 0.01 50 9 1991 -1.856 -3.585 to -0.127 NA NA NA NA
coin 50 0.01 100 9 1991 -2.815 -5.333 to -0.296 NA NA NA NA
coin 50 0.05 5 67 1933 -0.052 -0.331 to 0.226 NA NA -0.005019 -0.046354 to 0.036316
coin 50 0.05 10 67 1933 0.201 -0.204 to 0.606 NA NA 0.001035 -0.000399 to 0.002468
coin 50 0.05 20 67 1933 0.181 -0.397 to 0.759 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 50 0.05 50 67 1933 0.205 -0.650 to 1.060 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 50 0.05 100 67 1933 0.153 -1.031 to 1.338 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 100 0.01 5 15 1985 -0.506 -1.197 to 0.186 NA NA -0.168766 -0.371338 to 0.033807
coin 100 0.01 10 15 1985 -0.649 -1.306 to 0.008 NA NA 0.002015 0.000042 to 0.003988
coin 100 0.01 20 15 1985 -0.074 -0.986 to 0.838 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 100 0.01 50 15 1985 0.534 -1.235 to 2.303 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 100 0.01 100 15 1985 0.137 -2.026 to 2.301 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 100 0.05 5 61 1939 -0.014 -0.284 to 0.257 NA NA 0.009909 -0.022744 to 0.042562
coin 100 0.05 10 61 1939 0.040 -0.369 to 0.449 NA NA 0.000516 -0.000495 to 0.001526
coin 100 0.05 20 61 1939 -0.045 -0.624 to 0.534 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 100 0.05 50 61 1939 0.202 -0.744 to 1.148 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 100 0.05 100 61 1939 0.214 -0.977 to 1.405 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 200 0.01 5 19 1981 0.054 -0.382 to 0.491 NA NA 0.027259 0.020088 to 0.034430
coin 200 0.01 10 19 1981 0.756 0.068 to 1.444 NA NA 0.001514 -0.000198 to 0.003227
coin 200 0.01 20 19 1981 1.258 0.137 to 2.380 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 200 0.01 50 19 1981 2.982 1.684 to 4.279 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 200 0.01 100 19 1981 4.797 2.328 to 7.265 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 200 0.05 5 73 1927 0.021 -0.249 to 0.291 NA NA -0.005289 -0.051577 to 0.040999
coin 200 0.05 10 73 1927 0.146 -0.226 to 0.518 NA NA 0.001038 -0.000400 to 0.002476
coin 200 0.05 20 73 1927 0.185 -0.295 to 0.666 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 200 0.05 50 73 1927 0.742 -0.062 to 1.546 NA NA 0.000000 ≈0.000000 (degenerate CI)
coin 200 0.05 100 73 1927 1.359 0.144 to 2.574 NA NA 0.000000 ≈0.000000 (degenerate CI)

How to Cite This Page

Lucky Picks. “Research: Coin Toss Simulation — Hot, Cold & Independence.”
https://luckypicks.io/research/independence-and-lagging-numbers/coin-toss-simulation/
(Accessed September 2026)

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