Research: Coin Toss Simulation — Hot, Cold & Independence

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

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.

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

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.

Tertiary metric baseline: pHitWithin(2) = 0.746777 · MDE @80% power (|ΔpHit|) = 0.158331

Interpretation notes (important)

What these results do not show

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.