Research: Powerball Simulation — Hot, Cold & Independence

Generated: 2025-12-15T20:29:34.005Z · Lottery: Powerball (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 69
  • 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: 16
  • 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: 2.970

95% CI: -2.519 to 8.459

One-sided p (approx): 0.855525

n: lagging=35; control=1929

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: -0.235

95% CI: -0.441 to -0.029

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

Tertiary (bounded / UX metric): ΔpHitWithin(16) lagging − control

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

Estimate: -0.045812

95% CI: -0.204385 to 0.112761

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 = 2.970 (95% CI -2.519 to 8.459); one-sided p≈0.855525. Estimate is slightly later, 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): 13.402

n: lagging=35; control=1929

MDE @80% power(|ΔmeanWait|): 7.841

Secondary baseline meanCount@10: 0.721

MDE @80% power(|ΔmeanCount|): 0.294

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(16) = 0.702955 · MDE @80% power (|ΔpHit|) = 0.218225

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 1997 NA NA NA NA NA NA
target 50 0.01 10 0 1997 NA NA NA NA NA NA
target 50 0.01 20 0 1997 NA NA NA NA NA NA
target 50 0.01 50 0 1997 NA NA NA NA NA NA
target 50 0.01 100 0 1997 NA NA NA NA NA NA
target 50 0.05 5 50 1936 0.022 -0.165 to 0.208 0.148 -0.165 to 0.461 -0.011983 -0.140672 to 0.116705
target 50 0.05 10 50 1936 0.149 -0.116 to 0.414 -0.069 -0.913 to 0.776 0.074174 -0.063429 to 0.211776
target 50 0.05 20 50 1936 0.069 -0.258 to 0.396 -0.159 -2.094 to 1.775 0.003140 -0.113171 to 0.119452
target 50 0.05 50 50 1936 -0.155 -0.635 to 0.324 0.996 -2.869 to 4.860 -0.019855 -0.074532 to 0.034821
target 50 0.05 100 50 1936 -0.123 -0.937 to 0.692 0.891 -3.053 to 4.835 0.000000 ≈0.000000 (degenerate CI)
target 100 0.01 5 9 1986 0.304 -0.024 to 0.631 -0.779 -1.819 to 0.261 NA NA
target 100 0.01 10 9 1986 0.060 -0.231 to 0.350 -2.537 -4.728 to -0.346 NA NA
target 100 0.01 20 9 1986 -0.003 -0.812 to 0.806 -3.742 -8.673 to 1.189 NA NA
target 100 0.01 50 9 1986 0.261 -1.356 to 1.878 0.072 -13.013 to 13.157 NA NA
target 100 0.01 100 9 1986 0.859 -0.586 to 2.304 1.309 -13.861 to 16.480 NA NA
target 100 0.05 5 35 1929 -0.022 -0.183 to 0.140 -0.247 -0.735 to 0.241 0.030778 -0.127832 to 0.189388
target 100 0.05 10 35 1929 -0.235 -0.441 to -0.029 -0.122 -1.353 to 1.109 -0.106421 -0.271876 to 0.059034
target 100 0.05 20 35 1929 -0.338 -0.681 to 0.004 0.561 -1.976 to 3.099 -0.100185 -0.255071 to 0.054700
target 100 0.05 50 35 1929 -0.312 -0.868 to 0.243 2.584 -2.390 to 7.557 -0.037444 -0.114593 to 0.039706
target 100 0.05 100 35 1929 -0.338 -1.071 to 0.396 2.975 -2.514 to 8.464 0.000518 -0.000497 to 0.001534
target 200 0.01 5 5 1983 -0.176 -0.569 to 0.217 0.293 -0.493 to 1.079 NA NA
target 200 0.01 10 5 1983 0.257 -0.364 to 0.878 0.738 -1.822 to 3.297 NA NA
target 200 0.01 20 5 1983 0.949 -0.761 to 2.658 -0.624 -6.071 to 4.823 NA NA
target 200 0.01 50 5 1983 0.560 -1.710 to 2.830 -2.193 -9.466 to 5.081 NA NA
target 200 0.01 100 5 1983 1.162 -0.859 to 3.183 -2.428 -9.704 to 4.848 NA NA
target 200 0.05 5 35 1910 0.059 -0.160 to 0.277 0.122 -0.270 to 0.514 0.029245 -0.129383 to 0.187872
target 200 0.05 10 35 1910 -0.054 -0.308 to 0.200 0.157 -0.926 to 1.240 -0.022364 -0.189449 to 0.144722
target 200 0.05 20 35 1910 -0.047 -0.446 to 0.351 0.321 -2.004 to 2.646 -0.045625 -0.191576 to 0.100327
target 200 0.05 50 35 1910 -0.299 -0.931 to 0.334 0.711 -3.468 to 4.891 -0.011294 -0.066797 to 0.044209
target 200 0.05 100 35 1910 -0.640 -1.605 to 0.325 0.758 -3.732 to 5.247 0.000000 ≈0.000000 (degenerate CI)

How to Cite This Page

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

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