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.

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 = 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 December 2025)

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